Journal of Lie Theory Volume 9 (1999) 249–269 C ◯1999 Heldermann Verlag

Multi-Parameter Symmetries of First Order Ordinary Differential Equations

Sebastian Walcher

Communicated by P. J. Olver

Abstract. This is a collection of results on the use of infinitesimal orbital symmetries of first-order ordinary differential equations. Some of these results are classical, dating back to Lie and Bianchi, and some new results are added.

1. Introduction

Sophus Lie’s work on symmetries of differential equations (see [11], for instance) has been rediscovered and revived in recent years, and it forms the foundation for a wealth of new results. It seems that most of the recent work in the tradition of Lie has been focussed on partial differential equations or on ordinary differential equations of higher order. In contrast, this article deals exclusively with firstorder ordinary differential equations and their (infinitesimal orbital) symmetries. We present several results of Lie and his school, sometimes rephrased to fit the given situation, and add some new results.

We will consider autonomous, analytic ordinary differential equations ẋ = f(x), which are defined on an open and connected subset U of K^{n} (with K standing for the real or complex numbers). To every vector field f we assign the Lie derivative L_{f} which sends any scalar-valued function ψ to the function L_{f}(ψ) defined by L_{f}(ψ)(x) := Dψ(x)f(x). Specifically, we are interested in (orbital) symmetries of first order differential equations. As is well-known (see [17], for instance), an analytic vector field g (defined on some open subset Ũ of U) generates a local one-parameter group of orbital symmetries of such a differential equation (thus, the transformations map solution orbits of ẋ = f(x) to solution orbits, not necessarily preserving the parameterization) if and only if there is an analytic function λ such that [g, f] = λf on Ũ. (The Lie bracket of vector fields is defined as usual: [g, f] (x) = Df(x)g(x) − Dg(x)f(x). It follows that L_{[g,f]} = L_{g}L_{f} − L_{f}L_{g}.) In the case λ = 0 the parameterizations will also be preserved. Symmetries of a non-autonomous equation ẋ = h(t, x), as introduced by S. Lie, correspond to orbital symmetries of the “autonomized”

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag system ṫ = 1, ẋ = h(t, x). Moreover, f and μf, with μ analytic and μ(x) ≠ 0 for all x in an open subset Ũ of U, have the same orbits, hence the same orbital symmetries on any open subset of Ũ. In this sense, it is harmless to pass from f to μf. It should also be noted that orbital symmetries of ẋ = f(x) stand in correspondence with the symmetries of the linear, homogeneous partial differential equation L_{f}(ϕ) = 0. Several of Lie’s results (cf. Lie [11], Bianchi [2], Engel/Faber [3]) were originally stated for the latter type.

The vector fields g satisfying [g, f] = λf on some open Ũ ⊆ U form a Lie algebra but not necessarily a finite dimensional Lie algebra, and moreover, even a finite set of such vector fields will not, in general, generate a finite dimensional Lie algebra of vector fields. To illustrate this, consider the simple case f = (1, 0, …, 0)^{t}. Then

[g, f] = λf iff g(x) = ( \begin{matrix}γ(x) \\ ĝ(x_{2}, …, x_{n}) \end{matrix} ) ,

with ∂γ/∂x_{1} = λ. In general, due to the straightening theorem, a similar observation holds locally for any differential equation near a nonstationary point. Therefore, it would be unnatural to restrict attention to finite dimensional Lie algebras of infinitesimal orbital symmetries. (Things are different in the case of higher-order ordinary differential equations, where the infinitesimal symmetries automatically form a finite dimensional Lie algebra. This is due to a geometrically motivated restriction imposed on the admissible symmetries of the equivalent first-order system; a discussion can be found in Gaeta [4].)

The “multiparameter systems of infinitesimal orbital symmetries” to be discussed here are finite involution systems of vector fields generating local one-parameter groups of orbital symmetries. Recall that a set g_{1}, …, g_{r} of analytic vector fields defined on an open, connected subset U of K^{n} is called an involution system if there are analytic, scalar-valued functions μ_{ijk} on U such that [g_{i}, g_{j}] = ∑_{k} μ_{ijk}g_{k} for all i and j. Any finite set h_{1}, …, h_{s} of analytic vector fields on U can be extended to a (finite) set that is in involution on an open-dense subset U^{∗} of U. To see this, consider the h_{i} as elements of an n-dimensional vector space over the field L of all meromorphic functions on U. If some [h_{i}, h_{j}] is not a linear combination of h_{1}, …, h_{s} (equivalently, there is a point y in K^{n} such that [h_{i}, h_{j}] (y) is not a linear combination of h_{1}(y), …, h_{s}(y)), augment the given set by this vector field. Due to finite dimension over L, this process will terminate. (If convenient, it may also be assumed that the vector fields forming the involution system are linearly independent over L.) If, additionally, [h_{i}, f] = λ_{i}f for analytic λ_{i}, 1 ≤ i ≤ s, a similar property will also hold for the Lie brackets [h_{i}, h_{j}], and therefore for the involution system generated by the h_{i}.

Let us introduce a little more notation. For an open-dense U^{∗} ⊆ U, let N_{U^{∗}}(f) be the set of all vector fields g on U^{∗} such that [g, f] = λf for an analytic function λ on U^{∗}, and let C_{U^{∗}}(f) be the set of all vector fields g on U^{∗} such that [g, f] = 0. These will be called, respectively, the normalizer and centralizer of f on U^{∗}. Both the normalizer and the centralizer of f are Lie algebras over K, and also modules over the ring of first integrals of f on U^{∗}. (It is said that ψ is a first integral of f if L_{f}(ψ) = 0. The module property follows from the general rule [ψg, f] = ψ [g, f] − L_{f}(ψ)g.) Using this language, it may be said that this article is about subalgebras of N_{U^{∗}}(f), resp. C_{U^{∗}}(f) which are also finitely generated submodules. The exposition will include several classical results, due to Lie and Bianchi. (Sources for these are Lie [11], Engel/Faber [3], Bianchi [1], and Hermann [8], and more details can be found in these references. For a general overview of symmetries of differential equations we refer to Olver [12], [13]. A short account of Lie’s method for first-order ordinary differential equations in dimension two can also be found in Ince [9].) We will discuss the use of such systems of infinitesimal (orbital) symmetries to obtain nontrivial information about (and possibly simplify) ẋ = f(x).

Specifically, we will discuss the use of infinitesimal symmetries to find solution-preserving maps to equations of smaller dimension, and, ideally, to obtain solutions of ẋ = f(x) by “algebraic” operations and quadratures alone. (Since “integration” is an ambiguous term in our context, we will use the term “quadrature” to describe the determination of antiderivatives.) It has to be emphasized, though, that this is not the only useful application of symmetries. For instance, there is a large body of work (from various perspectives) on the influence that a compact linear symmetry group has on the qualitative behavior of a differential equation; see Golubitsky et al. [5], [6], and Scheurle [15], among others. Hadeler [7] discusses qualitative features of equations admitting the (non-compact) symmetry group R^{+} . Bluman and Kumei [2] briefly mention that local one-parameter groups of symmetries stabilize certain invariant sets, like separatrices of real equations in dimension two. As will be shown here, orbital symmetries enforce the existence of certain invariant sets for a differential equation under very general circumstances. (This result may be seen as a generalization of the familiar ”stratification of phase space” in the case of a compact linear symmetry group.) Therefore, even if Lie’s general local reduction procedure (which relies on non-constructive tools like the inverse function theorem) cannot be carried out for a given equation, one still gets nontrivial information in a relatively easy way. (Occasionally there will be a comment on the computational aspect of a general result, but we will not discuss this systematically.) A few examples and applications will be presented in the final section.

We will not discuss the (serious) problem of how to determine infinitesimal symmetries of first order ordinary differential equations. It is a regrettable fact that there is no general computational approach to these, and this distinguishes the class of first-order ordinary differential equations. We will usually assume that some symmetries are known (and note that it is frequently possible to start with a set of symmetries-to-be and then construct the symmetric vector fields). Most of the examples we discuss will therefore have a prescribed set of symmetries, although, as will be indicated, it is sometimes possible to find these symmetries in a constructive manner.

2. Reduction of dimension via invariants

Throughout this section let ẋ = f(x) be given on U, and vector fields g_{1}, …, g_{r} on U such that [g_{i}, f] = λ_{i}f, and [g_{i}, g_{j}] = ∑_{k} μ_{ijk}g_{k} for all i and j, with the λ_{i} and μ_{ijk} analytic on an open(-dense) U^{∗} ⊆ U. Denote by s the rank of g_{1}, …, g_{r} over the field L. In this section it will be assumed that s < n.

A well-known theorem of Frobenius states that there is an open and dense subset U^{∗∗} of U^{∗} and analytic functions ϕ_{1}, …, ϕ_{n−s} on U^{∗∗} that are independent (i.e., their functional matrix has rank n − s everywhere) common first integrals of g_{1}, …, g_{r}. Moreover, any point of U^{∗∗} has a neighborhood such that every common first integral of g_{1}, …, g_{r} in this neighborhood is a function of the ϕ_{j}. ¿From [g_{i}, f] = λ_{i}f it follows that L_{g_{i}}L_{f} − L_{f}L_{g_{i}} = λ_{i}L_{f}, and in particular L_{g_{i}}L_{f}(ϕ_{j}) = λ_{i}L_{f}(ϕ_{j}) for all i and j. This property is the basis of the following “reduction of dimension theorem”, which goes back to Lie [11]; see also Engel/Faber [3] and Hermann [8].

Proposition 2.1. Assume that [g_{i}, f] = 0 for i = 1, …, r.

(a) Then every point of U^{∗∗} has a neighborhood such that the restriction of Φ := (ϕ_{1}, …, ϕ_{n−s})^{t} is solution-preserving from ẋ = f(x) to an equation ẋ = h(x) on an open subset of K^{n−s}.

(b) Moreover, for every point of U^{∗∗} there is a neighborhood and analytic functions ϕ_{n−s+1}, …, ϕ_{n} such that Ψ := (ϕ_{1}, …, ϕ_{n})^{t} is invertible and solutionpreserving from ẋ = f(x) to an equation ẋ = f^{∗}(x), with

f^{∗}(x) = ( f_{1}^{∗}(x_{1}, …, x_{n−s}) ; ⋮ ; f_{n−s}^{∗}(x_{1}, …, x_{n−s}) ; f_{n−s+1}^{∗}(x_{1}, …, x_{n}) ; ⋮ ; f_{n}^{∗}(x_{1}, …, x_{n}) ) .

Here, the g_{i} are transformed to g_{i}^{∗} = (0, …, 0, ∗, …, ∗)^{t}, with zeros in the first n − s entries. Furthermore, one has [g_{i}^{∗}, f^{∗}] = 0, and the g_{i}^{∗} are locally in involution. Proof. (a) We have L_{g_{i}}L_{f} = L_{f}L_{g_{i}} for all i. Therefore, whenever ψ is a common first integral of the g_{i} then so is L_{f}(ψ), and hence it can be expressed as a function of the ϕ_{j} in a neighborhood of every point of U^{∗∗}. It follows that the entries of DΦ(x)f(x) are common first integrals of the g_{i}, and this shows the existence of h such that DΦ(x)f(x) = h(Φ(x)) for all x in this neighborhood. The proof of part (b), with arbitrary ϕ_{n−s+1}, …, ϕ_{n} such that ϕ_{1}, …, ϕ_{n} are functionally independent at the point in question, is similar. Since solutionpreserving maps preserve Lie brackets, the final assertions follow.

Remark 2.2. The results of (2.1), with the exception of the last statement, also hold for vector fields of the form f +∑ σ_{i}g_{i}, with analytic functions σ_{i} on U, and thus for a larger class of vector fields than those admitting the g_{i} as infinitesimal symmetries. Therefore, the proposition does not use all the information available from the symmetries.

To extend the procedure to the orbitally symmetric case (with the λ_{i} not necessarily zero), we use the following auxiliary result.

Lemma 2.3. Assume that [g_{i}, f] = λ_{i}f for i = 1, …, r, and let ψ be a common first integral of the g_{i} such that L_{f}(ψ) ≠ 0. Then

[g_{i}, \frac{1}{L_{f}(ψ)}f] = 0 for 1 ≤ i ≤ r.

Proof. The bracket conditions imply L_{g_{i}}L_{f}(ψ) = λ_{i}L_{f}(ψ), thus

L_{g_{i}}(L_{f}(ψ))^{−1} = −L_{f}(ψ)^{−2}L_{g_{i}}L_{f}(ψ) = −λ_{i}(L_{f}(ψ))^{−1},

and the assertion follows.

This transfer from normalizer to centralizer (on a smaller, but still open and dense subset of U where L_{f}(ψ) has no zeros) is not quite satisfactory in some respects. For instance, the zero set of L_{f}(ψ) contains all the stationary points of f, where interesting local behavior may occur.

The exceptional case that every common first integral of the g_{i} is also a first integral of f has been excluded in the lemma. This exceptional case occurs if and only if f is a linear combination of the g_{i} over L. (An equivalent characterization is that for all y in an open-dense subset of U, f(y) is a linear combination of g_{1}(y), …, g_{r}(y) over K.)

Proposition 2.4. Let the hypotheses and notation be as at the beginning of this section. Furthermore, assume that there is a common first integral ψ of g_{1}, …, g_{r} such that L_{f}(ψ) ≠ 0, and let Û := {z ∈ U^{∗∗} : L_{f}(ψ)(z) ≠ 0}.

(a) Then every point of Û has a neighborhood such that the restriction of Φ := (ϕ_{1}, …, ϕ_{n−s})^{t} is locally orbit-preserving from ẋ = f(x) to an equation ẋ = h(x) on an open subset of K^{n−s}.

(b) Moreover, for every point of Û there is a neighborhood and analytic functions ϕ_{n−s+1}, …, ϕ_{n} such that Ψ := (ϕ_{1}, …, ϕ_{n})^{t} is invertible and locally orbit-preserving from ẋ = f(x) to an equation ẋ = f^{∗}(x), with

f^{∗}(x) = ( f_{1}^{∗}(x_{1}, …, x_{n−s}) ; ⋮ ; f_{n−s}^{∗}(x_{1}, …, x_{n−s}) ; f_{n−s+1}^{∗}(x_{1}, …, x_{n}) ; ⋮ ; f_{n}^{∗}(x_{1}, …, x_{n}) ) .

The g_{i}^{∗} still satisfy [g_{i}^{∗}, f^{∗}] = λ_{i}^{∗}f^{∗}, with λ_{i}^{∗} = λ_{i} ∘ Ψ. Proof. This follows from (2.1) for (L_{f}(ψ))^{−1}f. (The cautious phrase “locally orbit-preserving” is only necessary for the complex case. If K is the real number field then Φ resp. Ψ map solution orbits of ẋ = f(x) on Û to solution orbits of the respective image equations.)

Note that the assertion remains true with h = 0, resp.

f^{∗} = (0, …, 0, ∗, …, ∗)^{t}

if every common first integral of the g_{i} is also a first integral of f, but then it does not provide any new information.

Remark 2.5. Finding common first integrals of the g_{i} may not be a simple matter, although the proof of Frobenius’ theorem (see Hermann [8]) is, up to some point, constructive provided that the vector fields in question can be integrated in an elementary manner. The following observation may be useful in the case that all [g_{i}, f] = 0: If ψ is a common first integral of the g_{i}, then the same is true for L_{f}(ψ), and repeated application of L_{f} to some nonempty set of common first integrals of g_{1}, …, g_{r} will produce an independent set ψ_{1}, …, ψ_{q} (with q ≤ s) of common first integrals such that every L_{f}(ψ_{j}) can locally be expressed as a function of ψ_{1}, …, ψ_{q}. It is obvious that a variant of (2.1) works in this situation. Thus, it is not necessary to know a maximal independent system of common first integrals from the start.

Under certain circumstances it is possible to carry out the reduction of dimension step-by-step. We only discuss the case of infinitesimal symmetries; the orbital symmetry case follows with (2.3).

Proposition 2.6. Assume that, in addition to the general hypotheses, [g_{i}, f] = 0 for all i, and furthermore that there is a p < r such that [g_{i}, g_{j}] = μ_{ij1}g_{1} + ⋯ + μ_{ijp}g_{p} for all i ≤ p and all j.

(a) Then g_{1}, …, g_{p} are in involution on U^{∗}. Let γ_{1}, …, γ_{n−q} (with q denoting the rank of g_{1}, …, g_{p} over L) be a functionally independent system of common first integrals of g_{1}, …, g_{p} on the open-dense subset U^{∗∗} of U^{∗}. Then every point of U^{∗∗} has a neighborhood such that every L_{g_{j}}(γ_{k}) is a function of γ_{1}, …, γ_{n−q} in this neighborhood.

(b) Let Γ : U^{∗∗} → K^{n−q} be defined by Γ(x) := (γ_{1}(x), …, γ_{n−q}(x))^{t}. Then there are analytic vector fields f̃, g̃_{p+1}, …, g̃_{r} on an open and dense subset Ũ of Γ(U^{∗∗}) ⊆ K^{n−q} such that DΓ(x)f(x) = f̃(Γ(x)), and DΓ(x)g_{i}(x) = g̃_{i}(Γ(x)) for p + 1 ≤ i ≤ r. Moreover, [g̃_{i}, f̃] = 0, and the g̃_{i} are in involution on an open and dense subset of Ũ. Proof. (a) The hypothesis implies in particular that g_{1}, …, g_{p} are in involution. For all i ≤ p, all j, and for 1 ≤ k ≤ n − q one has

L_{g_{i}}L_{g_{j}}(γ_{k}) = L_{g_{j}}L_{g_{i}}(γ_{k}) + μ_{ij1}L_{g_{1}}(γ_{k}) + … + μ_{ijp}L_{g_{p}}(γ_{k}) = 0,

thus L_{g_{i}}(γ_{k}) is again a first integral of g_{1}, …, g_{p}.

(b) The existence of f̃ and the g̃_{j} follows as in (2.1). Since solution-preserving maps respect Lie brackets, it is also clear that f̃ and the g̃_{j} commute. For j, l ≥ p + 1 one gets

\begin{aligned}[g̃_{j}, g̃_{l}] (Γ(x)) = DΓ(x) [g_{j}, g_{l}] (x) & = ∑_{m=1}^{r} μ_{jlm}(x)DΓ(x)g_{m}(x) \\ & = ∑_{m=p+1}^{r} μ_{jlm}(x)g̃_{m}(Γ(x)), \end{aligned}

as DΓ(x)g_{m}(x) = 0 for m ≤ p. Therefore, Γ being locally onto, every [g̃_{j}, g̃_{l}] (z) is a linear combination of the g̃_{m}(z) (with m > p) for all z in an open and dense subset of Ũ, and the g̃_{j} are in involution.

This opens up the possibility to employ common first integrals of the g̃_{j} in the next step, and thus obtain a stronger version of (2.1). One interesting special case for which (2.6) is applicable is when the g_{i} span a finite dimensional Lie algebra over K, and g_{1}, …, g_{p} span an ideal of this algebra. In this case, Bianchi [1] refined the result, using composition series of the Lie algebra. (One may, in general, have to modify the given involution system in order to apply (2.6). Note that “linear combinations” with first integrals of f as coefficients are allowed.)

Let us recall the reduction procedure in presence of a single infinitesimal orbital symmetry. If [g, f] = λf, and g is such that one knows (locally) a solution-preserving map Ψ from ẋ = (1, 0, …, 0)^{t} to ẋ = g(x), then this same map is solution-preserving from ẋ = f^{∗}(x) := μ(x)f̃(x_{2}, …, x_{n}) to ẋ = f(x), with ∂μ/∂x_{1} = λ. Since f̃ depends only on x_{2}, …, x_{n}, the solution of ẋ = f̃(x) amounts to solving an n − 1-dimensional differential equation and quadratures. From this every solution of ẋ = f^{∗}(x) is obtained by another quadrature. (Note that, due to the straightening theorem, a map Ψ with the desired properties exists locally near every nonstationary point of g. To carry out the procedure, however, one needs to know Ψ explicitly.) The above “reduction of dimension plus quadratures” has no counterpart in the multi-parameter situation, in general. There is, however, one special case worth mentioning:

Proposition 2.7. Let the hypotheses and notation be as at the beginning of this section, and assume furthermore that g_{1} ≠ 0 and [g_{1}, g_{j}] = μ_{1j1}g_{1} for all j. Let Ψ be an invertible analytic map from an open V ⊆ K^{n} to a neighborhood of some point of U^{∗} such that Ψ is solution-preserving from ẋ = (1, 0, …, 0) to ẋ = g_{1}(x).

Then there are analytic vector fields g_{2}^{∗}, …, g_{r}^{∗} and f^{∗} on V such that Ψ is solution-preserving from ẋ = f^{∗}(x) to ẋ = f(x), and from ẋ = g_{j}^{∗}(x) to ẋ = g_{j}(x) for all j. Moreover,

g_{j}^{∗}(x) = ( \begin{matrix}γ_{j}(x) \\ ĝ_{j}(x_{2}, …, x_{n}) \end{matrix} ) and f^{∗}(x) = μ(x) ( \begin{matrix}ϕ(x_{2}, …, x_{n}) \\ f̂(x_{2}, …, x_{n}) \end{matrix} ) .

The ĝ_{j} are in involution on a suitable open subset V̂ of K^{n−1} , and there are analytic λ̂_{j} on V̂ such that [ĝ_{j}, f̂] = λ̂_{j}f̂ for 2 ≤ j ≤ r.

Proof. The existence of the vector fields on V follows from invertibility of Ψ, and it was seen above that f^{∗} has the asserted form. Now [g_{1}, g_{j}] = μ_{1j1}g_{1} implies [g_{1}^{∗}, g_{j}^{∗}] = μ_{1j1}^{∗}g_{1}^{∗}, or ∂g_{j}^{∗}/∂x_{1} = μ_{1j1}^{∗} · (1, 0, …, 0)^{t}, whence entries number 2, …, n of g_{j}^{∗} are independent of x_{1} .

A simple computation shows that

[g_{j}^{∗}, g_{l}^{∗}] = ( \begin{matrix}⋆ \\ [ĝ_{j}, ĝ_{l}] \end{matrix} ) ,

with the Lie bracket in the last entry being taken in K^{n−1} , and it follows that the ĝ_{j} are in involution. Similarly, it follows that [ĝ_{j}, f̂] = λ̂_{j}f̂ if μ has no zeros in a neighborhood of the point under consideration.

Thus, g_{1} reduces the solution of ẋ = f(x) to solving a differential equation in dimension n − 1, plus quadratures, and the remaining symmetries from g_{2}, …, g_{r} survive for the reduced differential equation. This latter property is the important one. The following consequence (when applied to finite-dimensional solvable Lie algebras) is sometimes called “Bianchi’s theorem”.

Corollary 2.8. In addition to the hypotheses of (2.7), assume that

[g_{2}, g_{j}] =  μ_{2j1}g_{1} + μ_{2j2}g_{2}
⋮  ⋮
[g_{r−1}, g_{j}] = μ_{r−1,j,1}g_{1} + … + μ_{r−1,j,r−1}g_{r−1}

for all j, and that g_{1}, …, g_{r} has rank r over L.

Then solving ẋ = f(x) can (locally) be reduced to solving an (n − r)- dimensional differential equation and quadratures.

It seems that Bianchi formulated this only for finite dimensional solvable Lie algebras, although in other parts of [1] he also considered general involution systems. The proper formulation of Bianchi’s theorem for higher-order differential equations was first given by Olver; cf. [12], Thm. 2.64. (We remark that whenever the condition on the orbit dimensions of the prolongated actions in this theorem is not satisfied then one necessarily gets a nontrivial first integral by (4.2), which also allows a reduction of the order.)

Even if the vector fields at hand can be explicitly integrated, it may be troublesome or even impossible to implement any of the reduction procedures discussed in this section. The implicit function theorem and the inverse function theorem are the basis for many of the results presented here, and these are not constructive. One may be able to actually carry out a variant of the reduction from (2.1), for instance, if all the functions and vector fields involved are polynomial, and one uses a more subtle approach. For (compact) linear symmetry groups, this procedure is quite familiar, see Rumberger/Scheurle [14] and Scheurle [15], for instance, where the symmetric vector fields are quite general, and qualitative features are of primary interest. (In the case of compact linear symmetry groups qualitative investigations are greatly facilitated by the fact that the reducing map is proper.) Hadeler [7] discusses qualitative features of homogeneous vector fields, with the (non-compact) symmetry group R^{+} . The case of orbital linear symmetries is dicussed in [10].

3. Invariant sets

This section is motivated by the well-known strategy to find group-invariant solutions (cf. Olver [12]), and also by the “stratification of the phase space” of differential equations admitting a (compact) linear symmetry group, see Gaeta [4], Golubitsky et al. [5], [6], and Scheurle [15].

In this section it will be assumed that f and g_{1}, …, g_{r} are analytic vector fields on U, and that [g_{i}, f] = λ_{i}f with analytic functions λ_{i} on U. (We do not assume that the g_{i} are in involution.) The basic question we are interested in is the following: What invariant sets does ẋ = f(x) necessarily admit as a consequence of the existence of the g_{i}?

One answer is the following.

Theorem 3.1. (a) The set

Y := {y ∈ U : f(y), g_{1}(y), …, g_{r}(y) are linearly dependent in K^{n}}

is invariant for ẋ = f(x).

(b) If, in addition, all [g_{i}, f] = 0 then

Z := {z ∈ U : g_{1}(y), …, g_{r}(y) are linearly dependent in K^{n}}

is invariant for ẋ = f(x).

Proof. (i) We will use the following invariance criterion (see [17], for instance): Let ρ_{i} and σ_{ij} (1 ≤ i, j ≤ q) be analytic on U, and L_{f}(ρ_{i}) = ∑_{j} σ_{ij}ρ_{j} for all i. Then the set of common zeros of ρ_{1}, …, ρ_{q} is invariant for ẋ = f(x).

(To see this, let v(t) be a solution of ẋ = f(x). Then w(t) := (ρ_{1}(v(t)), …, ρ_{q}(v(t)))^{t} solves a homogeneous linear differential equation. Thus, w(0) = 0 implies w(t) = 0 for all t.) (ii) We also need an auxiliary result from (multi-)linear algebra. Denote by ν_{1}, …, ν_{n} the dual basis of K^{n}, thus ν_{i} ((x_{1}, …, x_{n})^{t}) = x_{i}, and fix some q ≤ n.

Let 1 ≤ i_{0} < … < i_{q} ≤ n be a strictly increasing sequence of integers, and consider the map

Δ_{i_{0},...,i_{q}} : (K^{n})^{q+1} → K, Δ_{i_{0},...,i_{q}}(v_{0}, …, v_{q}) = det (ν_{i_{k}}(v_{l}))_{k,l} .

This map is multilinear and alternating, and (by the universal property of the Grassmann algebra) every multilinear and alternating map from (K^{n})^{q+1} to K is a linear combination of such Δ_{i_{0},...,i_{q}}.

Now let B : K^{n} → K^{n} be a linear map, and j_{0} < … < j_{q} fixed. Then it is easy to verify that

δ_{B} : (v_{0}, …, v_{q}) ↦ ∑_{i=0}^{q} Δ_{j_{0},...,j_{q}} (v_{0}, …, v_{i−1}, Bv_{i}, v_{i+1}, …, v_{q})

is multilinear and alternating, and therefore a linear combination of the Δ_{i_{0},...,i_{q}}. Moreover, the coefficients are linear functions of the entries of B.

(iii) We now prove part (a). Note that Y is the common zero set of all ρ_{i_{0},...,i_{r}}(x) := Δ_{i_{0},...,i_{r}} (f(x), g_{1}(x), …, g_{r}(x)). Fix j_{0} < … < j_{r}, and abbreviate Δ := Δ_{j_{0},...,j_{r}}, and ρ := ρ_{j_{0},...,j_{r}}. By multilinearity of the determinant, the Lie derivative of ρ is as follows:

L_{f}(ρ)(x) = Dρ(x)f(x) = Δ (Df(x)f(x), g_{1}(x), …, g_{r}(x))

+ ∑_{i} Δ (f(x), g_{1}(x), …, g_{i−1}(x), Dg_{i}(x)f(x), g_{i+1}(x), …, g_{r}(x))

= Δ (Df(x)f(x), g_{1}(x), …, g_{r}(x))

+ ∑_{i} Δ (f(x), g_{1}(x), …, g_{i−1}(x), Df(x)g_{i}(x), g_{i+1}(x), …, g_{r}(x))

− ∑_{i} Δ (f(x), g_{1}(x), …, g_{i−1}(x), λ_{i}(x)f(x), g_{i+1}(x), …, g_{r}(x)) ,

since [g_{i}, f] = λ_{i}f. The last term vanishes, and the remaining sum can, according to (ii), be expressed as a linear combination of the ρ_{i_{0},...,i_{r}}, with the coefficients being linear in the entries of Df(x), and hence analytic on U. Now the criterion from (i) applies. (iv) The proof of (b) is an easier variant of (iii).

Since any intersection, or set-theoretic difference, of invariant sets is invariant, we get

Corollary 3.2. Let q ≤ r. In the situation of (3.1 a), the set

Y_{q} := {y ∈ U : rank (f(y), g_{1}(y), …, g_{r}(y)) = q + 1}

is invariant for ẋ = f(x). In the situation of (3.1 b), the set

Z_{q} := {z ∈ U : rank (g_{1}(y), …, g_{r}(y)) = q}

is invariant for ẋ = f(x).

If λ_{i} ≠ 0 then Z is not, in general, invariant for ẋ = f(x). A simple example is given by g(x) = x and f a nonzero constant. Here we have [g, f] = −f, while Z = {0} is not invariant for ẋ = f(x). To illustrate that (3.1) can be useful, consider

f(x) := ( \begin{matrix}x_{1}^{2} − x_{2}x_{3} \\ 2x_{1}x_{2} \\ 2x_{1}x_{3} \end{matrix} ) , g(x) := ( \begin{matrix}x_{1} \\ 3x_{2} \\ −x_{3} \end{matrix} ) , and h(x) := ( \begin{matrix}x_{1}x_{2} \\ x_{2}^{2} \\ −x_{1}^{2} \end{matrix} ) .

It is easy to verify that g and h normalize f, and therefore the set of zeros of

ρ(x) := det (f(x), g(x), h(x)) = −x_{2} (x_{1}^{2} + x_{2}x_{3})^{2}

is invariant for ẋ = f(x), due to (3.1). Thus we have found the invariant plane x_{2} = 0 and the (less obvious) invariant cone x_{1}^{2} + x_{2}x_{3} = 0. Application of (3.1) to f and g alone turns out to produce the invariant straight lines x_{1} = x_{2} = 0, x_{1} = x_{3} = 0, and x_{2} = x_{3} = 0, and so on.

In the recent article [16] by Ünal, the result of (3.1) is proven (from a somewhat different perspective) in the special case r = n − 1 (see [16], Theorem 1), and some examples are given.

The following may also be worth recording.

Remark 3.3. If the g_{i} are in involution on U^{∗} ⊆ U, then the sets Y_{q} ∩ U^{∗} and Z_{q} ∩ U^{∗} are invariant for all g_{i}. The proof of this is similar to the one of (3.1) and (3.2).

Let us briefly return to the situation of §2, assuming that the g_{i} are analytic on U, and the μ_{ijk} analytic on U^{∗}. Then U \ U^{∗} is contained in Z, as an application of Cramer’s rule over L shows. Thus, if all λ_{i} = 0 then (2.1) takes care of the points in U^{∗} while (3.1) and (3.2) take care of Z (and there are local coordinates on Z_{q} \ Z_{q−1} which allow, in principle, to transfer the restriction of ẋ = f(x) to an open subset of K^{q}). Thus, the points in U \ U^{∗} are actually quite interesting. Moreover, the equations of the invariant sets determined so far can actually be computed.

The following result works for arbitrary solution-preserving maps, whether they are determined from symmetries or not.

Proposition 3.4. Let the analytic differential equation ẋ = h(x) be defined on V ⊆ K^{m}, and Φ : U → V be solution-preserving from ẋ = f(x) to ẋ = h(x). Then every set W_{q} := {x ∈ U : rank DΦ(x) = q} is invariant for ẋ = f(x).

Proof. Denote by F (t, y) (resp. H(t, y)) the solution of ẋ = f(x), x(0) = y (resp. ẋ = h(x), h(0) = y). Then F and H are analytic, and, by definition, Φ(F (t, y)) = H(t, Φ(y)). Differentiate this with respect to y to obtain

DΦ(F (t, y)) D_{2}F (t, y) = D_{2}H(t, Φ(y)) DΦ(y),

with D_{2} symbolizing the partial derivative with respect to the second variable. Now, D_{2}F (t, y) solves a linear differential equation with the identity matrix as initial value, and hence is always invertible, and a similar observation holds for D_{2}H. Therefore, the rank of DΦ(F (t, y)) is equal to the rank of DΦ(y) for all t.

A look at (2.1)ff. may give the impression that this is not a useful result, since the rank of Φ is, by construction, constant on the set under consideration. But in fact one often obtains nontrivial information. For instance, assume that U ⊆ K^{4} and that Φ(x) := ({x_{1}^{2}+x_{2}^{2} \atop x_{3}^{2}+x_{4}^{2}} ) is solution-preserving from ẋ = f(x) to some equation ẋ = h(x). Then the intersections of U with the planes defined by x_{1} = x_{2} = 0, resp. x_{3} = x_{4} = 0 are invariant for ẋ = f(x), as the criterion from (3.4) shows. (Such a situation occurs, for instance, if f is so(2) × so(2)-symmetric, but there are more general equations admitting this solution-preserving map; see (2.2).) It should also be noted that in the case of a finite linear symmetry group, with Φ constructed from a set of generators of the invariant algebra, (3.4) yields all the strata.

To finish this section, let us investigate how first integrals of f behave in presence of the g_{i}. As will be seen, this may yield an approach to finding new invariant sets from given ones.

Proposition 3.5. (a) If ψ is a first integral of f on U then so is every L_{g_{i}}(ψ).

(b) Any functionally independent set of first integrals of f (on an open-dense subset of U) can be extended to a functionally independent set ψ_{1}, …, ψ_{p} of first integrals of f such that every L_{g_{i}}(ψ_{j}) is locally a function of the ψ_{l}.

Proof. Part (a) is a consequence of L_{g_{i}}L_{f} − L_{f}L_{g_{i}} = λ_{i}L_{f}, while part (b) follows from repeatedly applying the L_{g_{i}} to the given set.

Corollary 3.6. Let the hypotheses and ψ_{1}, …, ψ_{p} be as in (3.5) on the open and dense subset U^{∗} of U, with p ≥ 1.

(a) Then for every point of U^{∗} there is a neighborhood and functions ψ_{p+1}, …, ψ_{n} such that Ψ := (ψ_{1}, …, ψ_{n})^{t} is invertible on this neighborhood. There are vector fields f^{∗}, g_{1}^{∗}, …, g_{r}^{∗} such that DΨ(x)f(x) = f^{∗}(Ψ(x)), and DΨ(x)g_{i}(x) = g_{i}^{∗}(Ψ(x)). Moreover,

f^{∗}(x) = ( 0 ; ⋮ ; 0 ; f̂(x) ) , and g_{i}^{∗}(x) = ( γ_{i,1}(x_{1}, …, x_{p}) ; ⋮ ; γ_{i,p}(x_{1}, …, x_{p}) ; g̃_{i}(x) ) .

(b) There is a q ≤ p (actually, q is the rank of the matrix (γ_{i,k}(x_{1}, …, x_{p})) in an open and dense subset V of the given neighborhood) and linear combinations g_{q+1}^{∗∗}, …, g_{r}^{∗∗} of the g_{i}^{∗} (with analytic first integrals of f as coefficients) such that

g_{j}^{∗∗}(x) = ( 0 ; ⋮ ; 0 ; ĝ_{j}(x) ) for j > q.

(c) Fix z ∈ V . Then

\begin{matrix}[f̂(z_{1}, …, z_{p}, x_{p+1}, …, x_{n}), ĝ_{j}(z_{1}, …, z_{p}, x_{p+1}, …, x_{n})] \\ = λ̂_{j}(x_{p+1}, …, x_{n})f̂(z_{1}, …, z_{p}, x_{p+1}, …, x_{n}) \end{matrix}

(the bracket is taken in K^{n−q}) for all j > q. Moreover, if the g_{i} are in involution then the ĝ_{j}(z_{1}, …, z_{p}, x_{p+1}, …, x_{n}) are in involution.

Proof. In part (a), the existence of the ψ_{l} (with l > p) is clear, as is the existence of the g_{i}^{∗} and of f^{∗}. The special form of f^{∗} follows from ψ_{1}, …, ψ_{p} being first integrals, while the special form of the g_{i}^{∗} is a consequence of (3.5). Part (b) follows with Gauss’ algorithm (over L) applied to the g_{i}^{∗}. (In case γ_{1,1} ≠ 0 the first step is to change g_{j}^{∗} to g_{j}^{∗} − (γ_{j,1}/γ_{1,1})g_{1}^{∗} (j > 1) to create zeros in the first entry. Note that the coefficients are indeed first integrals of f.) Part (c) follows from simple computations.

There is the usual problem with this proposition: It may be impossible to carry out the necessary coordinate transformations explicitly. A weaker version, which avoids this problem, can be stated as follows: Whenever σ is a first integral of f and, for instance, L_{g_{1}}(σ) ≠ 0, then g̃_{j} := L_{g_{1}}(σ)g_{j} − L_{g_{j}}(σ)g_{1} (for j ≥ 2) is also contained in the normalizer of f and admits the first integral σ. Moreover the g̃_{j} are in involution on an open and dense subset of U if the g_{i} are in involution on an open and dense subset of U. In this sense, there are infinitesimal symmetries of f that allow restriction to the level sets of σ. Obviously, this process can be continued.

4. First integrals from involution systems

At the end of §3 it has been shown how given first integrals of f can be put to use in presence of infinitesimal symmetries. In this section we will see that first integrals of f may actually be found from an involution system of infinitesimal symmetries.

Throughout this section we will assume that f, g_{1}, …, g_{r} are analytic vector fields on U, that [g_{i}, f] = λ_{i}f for all i, with analytic λ_{i}, that [g_{i}, g_{j}] = ∑_{k} μ_{ijk}g_{k}, with analytic μ_{ijk}, on an open and dense subset U^{∗} of U, and that g_{1}, …, g_{r} are linearly independent over L.

We will distinguish two cases. The first (which will be referred to as “nondegenerate”) is characterized by the condition that f, g_{1}, …, g_{r} are linearly independent over L, and its discussion goes back to Lie [11]; see the exposition in Engel/Faber [3]. The second (“degenerate”) case is that f, g_{1}, …, g_{r} are linearly dependent over L, and it seems that it was not discussed so far. (From the point of view of linear partial differential equations this is understandable.) In any case, it will turn out that there is a systematic reduction to the case of finite dimensional Lie algebras of infinitesimal (orbital) symmetries, by way of first integrals.

Proposition 4.1. In the nondegenerate case, the following holds:

(a) The μ_{ijk} are first integrals of f.

(b) If not all the μ_{ijk} are constant then there exist functionally independent first integrals ϕ_{1}, …, ϕ_{q} (with q > 0) on an open-dense U^{∗∗} ⊆ U^{∗} with the following property: For each point of U^{∗∗} there is a neighborhood and analytic functions ϕ_{q+1}, …, ϕ_{n} such that Φ := (ϕ_{1}, …, ϕ_{n})^{t} is invertible on this neighborhood, and solution-preserving from ẋ = f(x) to ẋ = f^{∗}(x), and from ẋ = g_{i}(x) to ẋ = g_{i}^{∗}(x), 1 ≤ i ≤ r. Moreover,

f^{∗}(x) = ( 0 ; ⋮ ; 0 ; f̂(x) ) and g_{i}^{∗}(x) = ( γ_{i,1}(x_{1}, …, x_{q}) ; ⋮ ; γ_{i,q}(x_{1}, …, x_{q}) ; g̃_{i}(x) ) ,

and [g_{i}^{∗}, g_{j}^{∗}] = ∑_{k} μ_{ijk}^{∗}g_{k}^{∗}, with the μ_{ijk}^{∗} functions of x_{1}, …, x_{q}. Furthermore, there is an integer p ≤ q and linear combinations g_{j}^{∗∗} of the g_{i}^{∗} (j > q), with the coefficients being functions of x_{1}, …, x_{q}, and

g_{j}^{∗∗}(x) = ( 0 ; ⋮ ; 0 ; ĝ_{j}(x) ) .

(c) Fix z in this neighborhood. Then the ĝ_{j}(z_{1}, …, z_{q}, x_{q+1}, … x_{n}) form a finite dimensional Lie algebra of infinitesimal orbital symmetries for

f̂(z_{1}, …, z_{q}, x_{q+1}, … x_{n})

on a suitable open subset of K^{n−q}.

Proof. As to part (a), we have [[g_{i}, g_{j}] , f] = α_{ij}f, with suitable α_{ij}, from the Jacobi identity, and [[g_{i}, g_{j}] , f] = ∑_{k} [μ_{ijk}g_{k}, f] = ∑_{k} μ_{ijk}λ_{k}f −∑_{k} L_{f}(μ_{ijk})g_{k}. The linear independence of f, g_{1}, …, g_{r} over L now shows L_{f}(μ_{ijk}) = 0. Parts (b) and (c) follow from an application of (3.6) to a maximal functionally independent subset of the μ_{ijk}. The assertion about finite dimension in (c) follows from the fact that the coefficients occurring in the involution systems are functions of x_{1}, …, x_{q}.

Thus, a “proper” involution system may provide more readily accessible information about ẋ = f(x) than a finite dimensional Lie algebra of infinitesimal symmetries. Actually, the argument of (4.1) is also useful for certain finite dimensional Lie algebras.

Remark 4.2. The dimension of a finite dimensional Lie algebra of vector fields on U is greater than or equal to the dimension over L of the corresponding involution system. The latter is equal to the “generic” dimension of the local group orbits, and it is equal to the dimension of the Lie algebra if and only if the local group action is semi-regular on an open and dense subset of U. In all other cases, (4.1) yields nontrivial first integrals.

For a specific example, consider the vector fields

g_{1}(x) = ( \begin{matrix}0 \\ −x_{3} \\ x_{2} \\ 0 \end{matrix} ) and g_{2}(x) = ( \begin{matrix}0 \\ 0 \\ −x_{4} \\ x_{3} \end{matrix} )

on K^{4} . These generate a three-dimensional Lie algebra (isomorphic to so(3)), and the corresponding involution system has dimension 2 over L, as follows from [g_{1}, g_{2}] = \frac{x_{4}}{x_{3}} g_{1} + \frac{x_{2}}{x_{3}} g_{2} . Therefore, every vector field f that is defined on an open subset U of K^{4} and admits the infinitesimal orbital symmetries g_{1} and g_{2} also admits the first integrals ρ_{1}(x) = \frac{x_{4}}{x_{3}} and ρ_{2}(x) = \frac{x_{2}}{x_{3}} , provided that the nondegeneracy condition holds. It follows that

f(x) = ( \begin{matrix}ϕ(x) \\ μ(x)x_{2} \\ μ(x)x_{3} \\ μ(x)x_{4} \end{matrix} ) = μ(x) ( \begin{matrix}ψ(x) \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix} ) ,

with analytic functions μ, ϕ and ψ on an open-dense subset of U. (A simple computation shows that the g_{i} are infinitesimal orbital symmetries if and only if ψ is a common first integral of the g_{i}.) There are other methods to obtain this result, in particular since the g_{i} are linear vector fields, but the method from (4.1) seems straightforward and useful in quite general circumstances.

Now let us turn to the degenerate case. It follows from the general hypothesis of this section that there are analytic functions β_{1}, …, β_{r} on an open and dense subset U^{∗} of U such that f = ∑_{i} β_{i}g_{i}. (We will assume that f ≠ 0.) The following can be said in this situation.

Proposition 4.3. (a) Assume that β_{1} ≠ 0. Then all β_{i}/β_{1} (i > 1) are first integrals of f, and passage to the orbit-equivalent vector field f̃:= β_{1}^{−1}f yields f̃ = ∑_{i} β̃_{i}g_{i}, with all β̃_{i} first integrals of f̃. If all λ_{i} = 0 then all β_{i} are first integrals of f.

(b) If all β_{i} are first integrals of f then all β_{l}μ_{ijk} − β_{k}μ_{ijl} are first integrals of f.

Proof. Part (a) follows from 0 = [f, f] = ∑ [f, β_{i}g_{i}] = −αf + ∑ L_{f}(β_{i})g_{i}, with α := ∑ β_{i}λ_{i}. The linear independence of the g_{i} over L forces L_{f}(β_{i}) = αβ_{i} for all i, and from this the assertion follows. Part (b) follows similarly: There are analytic functions λ_{ij} such that [[g_{i}, g_{j}] , f] = λ_{ij}f; on the other hand [[g_{i}, g_{j}] , f] = ∑ [μ_{ijk}g_{k}, f] = (∑ μ_{ijk}λ_{k})f − ∑ L_{f}(μ_{ijk})g_{k}, whence L_{f}(μ_{ijk}) = ν_{ij}β_{k}, and L_{f}(β_{l}μ_{ijk}) = ν_{ij}β_{k}β_{l}.

This gives rise to a (rather obvious) application of (3.6), which will not be written down explicitly, but a brief look at the case when no nonconstant first integrals can be gained from this procedure may be in order. First, we may assume that all β_{i} are constant, and then there is no loss of generality in assuming that f = g_{1} . Part (b) then shows that all μ_{ijk} with k > 1 are constants.

In a sense, it is possible to reduce the degenerate to the nondegenerate case, as the following shows. For a given set of vector fields g_{i}, however, this may not be the most advisable strategy.

Proposition 4.4. Let the hypotheses and notation be as in (4.3). Let ϕ be analytic on U such that L_{f}(ϕ) ≠ 0, and define g̃_{i} := g_{i} − (L_{g_{i}}(ϕ)/L_{f}(ϕ)) f. Then the system g̃_{1}, …, g̃_{r} has rank r − 1 over L, and [g̃_{i}, f] = λ̃_{i}f for suitable analytic λ̃_{i}, 1 ≤ i ≤ r.

If g̃_{2}, …, g̃_{r} are linearly independent over L then they are in involution. Moreover, there is a nonzero analytic function σ on an open-dense subset of U such that [g̃_{i}, σf] = 0 for i > 1.

Proof. By construction, the g̃_{i} have the common first integral ϕ, and [g̃_{i}, f] = λ̃_{i}f. If g̃_{2}, …, g̃_{r} are linearly independent over L then f, g̃_{2}, …, g̃_{r} are linearly independent over L, since a relation f = ∑_{i>1} β̃_{i}g̃_{i} contradicts L_{f}(ϕ) ≠ 0. Hence f, g̃_{2}, …, g̃_{r} are in involution, and therefore, for all i, j ≥ 2 there are ν̃_{ij} and μ̃_{ijk} such that [g̃_{i}, g̃_{j}] = ν̃_{ij}f + ∑_{k>1} μ̃_{ijk}g̃_{k}, and L_{[g̃_{i},g̃_{j}]}(ϕ) = 0 shows that ν̃_{ij} = 0. The last assertion, with σ = 1/L_{f}(ϕ), follows from (2.3).

In case f = g_{1} , elementary linear algebra shows that μ̃_{ijk} = μ_{ijk} whenever i, j, k ≥ 2.

5. Some examples and applications

5.1. Equations in dimension three with two infinitesimal symmetries. It is well-known that knowledge of a nontrivial infinitesimal (orbital) symmetry g of an equation ẋ = f(x) on U ⊆ K^{2} reduces the determination of a first integral to a quadrature, since an integrating factor can be constructed from g; cf. Olver [12], Theorem 2.48. (Furthermore, up to determining the inverse of a certain transformation, the solution of ẋ = f(x) is thus reduced to quadratures.) Here we will discuss the case of an equation ẋ = f(x) on U ⊆ K^{3} , with two given normalizer elements g, h. We will always assume that f, g and h are linearly independent over L. (This excludes pathological cases like h = g + ρf, which should not qualify as a second nontrivial infinitesimal symmetry.) We will take the traditional point of view that ẋ = f(x) is ”solved” as soon as there are two independent first integrals, and that necessary coordinate transformations do not cause problems. It seems unnatural to assume a priori that g and h are in involution, and therefore we will not do so. The discussion of the degenerate case in §4 turns out to be quite useful here.

By Cramer’s rule over L, there are analytic α, σ and τ on an open and dense subset of U such that [g, h] = αf + σg + τh. Then (4.3 b) shows that L_{f}(σ) = L_{f}(τ) = 0, and it was shown in (3.5) that applying L_{g} or L_{h} to a first integral of f will again produce a first integral of f. Now there are three cases to be distinguished.

In the first case, there are two functionally independent first integrals of f among σ, τ, L_{g}(σ), L_{g}(τ), L_{h}(σ), L_{h}(τ). (If there are less than two in this list, then further applications of L_{g} or L_{h} are to no avail.) In this case, we consider ẋ = f(x) as essentially solved.

In the second case the above list contains one, and only one, independent first integral of f, say σ, and g̃ := L_{g}(σ)h−L_{h}(σ)g is a nonzero vector field that normalizes f, yet is not a multiple of f over L. (This would only be possible if L_{g}(σ) = L_{h}(σ) = 0, a contradiction to the linear independence of f, g and h over L.) In appropriate local coordinates one has

f = ( 0 ; f̂) and g = ( 0 ; ĝ ) ,

and the problem has been reduced to dimension 2.

The third case is characterized by σ = const. and τ = const. Consequently, if one defines J := {μf : μ analytic}, then (K · g + K · h + J) /J is a two-dimensional Lie algebra over K, and we can assume that σ = 0, τ = ϵ, with ϵ ∈ {0, 1}.

Now assume, with no loss of generality, that x_{1} is not a first integral of f, and define g̃ := g − (L_{g}(x_{1})/L_{f}(x_{1})) f, and h̃ := h − (L_{h}(x_{1})/L_{f}(x_{1})) f, according to (4.4). Then g̃ and h̃ normalize f, and [g̃, h̃] = ϵh̃, as follows from (4.4) and the observation thereafter. Moreover, we may assume (by going to (1/L_{f}(x_{1}))f, if necessary) that [g̃, f] = [h̃, f] = 0. Note that L_{f}(x_{1}) is, by (2.5), a common first integral of g and h, and thus a function of x_{1} alone. (If the normalization according to (2.3) has been carried out then one even has L_{f}(x_{1}) = 1.) Thus far we have (dropping the tilde)

g = ( 0 ; ĝ ) , h = ( 0 ; ĥ ) , f(x) = ( \begin{matrix}ϕ_{1}(x_{1}) \\ ϕ_{2}(x) \\ ϕ_{3}(x) \end{matrix} ) .

As far as ĝ and ĥ are concerned, this is a two-dimensional system, with x_{1} as a parameter, and for every fixed value of x_{1} one has [ĝ, ĥ] = ϵh. Here 1/ det(ĝ, ĥ)

is an integrating factor for ĥ, and thus determining a first integral for ĥ is a quadrature problem. This is also a first integral for h, and it is necessarily functionally independent from x_{1} . Thus a suitable coordinate transformation leads to h = (0, 0, ∗)^{t}, and then to h = (0, 0, 1)^{t}. (These transformations can be chosen so that they fix the first coordinate. One should use a new name for the transformed h, but there is little danger of confusion.) Using [h, f] = 0 and [g, h] = ϵh, one obtains

f = ( \begin{matrix}ϕ_{1}(x_{1}) \\ ϕ_{2}(x_{1}, x_{2}) \\ ϕ_{3}(x_{1}, x_{2}) \end{matrix} ) , g = ( \begin{matrix}0 \\ γ_{2}(x_{1}, x_{2}) \\ γ_{3}(x) \end{matrix} ) , h = ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ) .

Note that solving ẋ_{3} = ϕ_{3}(x_{1}, x_{2}) requires only quadratures once x_{1} and x_{2} are determined. Now [g, f] = 0 forces

[( \begin{matrix}ϕ_{1}(x_{1}) \\ ϕ_{2}(x_{1}, x_{2}) \end{matrix} ) , ( \begin{matrix}0 \\ γ_{2}(x_{1}, x_{2}) \end{matrix} )] = 0,

and this is, once again, a two-dimensional problem. Thus, up to quadratures and coordinate transformations, the equation ẋ = f(x) has been solved.

5.2. Involution systems of rank 2. The simplest nontrivial case of multiparameter symmetries occurs when ẋ = f(x) admits two infinitesimal orbital symmetries g and h on U ⊆ K^{n} such that f, g and h are linearly independent over L, and g and h are in involution. (In particular, n ≥ 3.) Therefore we will take a closer look at involution systems g, h such that g and h are linearly independent over L, and [g, h] = σg + τh. Locally, near any point y such that g(y) and h(y) are linearly independent, there are n − 2 common first integrals (due to Frobenius’ theorem), which, after a coordinate transformation, may be taken as x_{3}, …, x_{n}. Moreover, h can be straightened, hence there is no loss of generality in assuming that

h = ( 1 0 0 ⋮ 0 ) , g = ( γ_{1} ; γ_{2} ; 0 ; ⋮ ; 0 ) , and [g, h] = − ( ∂γ_{1}/∂x_{1} ; ∂γ_{2}/∂x_{2} ; 0 ; ⋮ ; 0 ) ,

with γ_{2} ≠ 0, and

σ = −\frac{∂γ_{2}/∂x_{1}}{γ_{2}} , τ = \frac{(∂γ_{2}/∂x_{1}) γ_{1} − (∂γ_{1}/∂x_{1}) γ_{2}}{γ_{2}} .

It is sometimes useful to further normalize g, while leaving h and the common first integrals intact. The admissible transformations are of the form

Ψ(x) = ( x_{1} + ψ_{1}(x_{2}, …, x_{n}) ; ψ_{2}(x_{2}, …, x_{n}) ; ψ_{3}(x_{3}, …, x_{n}) ; ⋮ ; ψ_{n}(x_{3}, …, x_{n}) ) ,

as is easily verified, and one has DΨ(x)g(x) = ĝ(Ψ(x)), with the entries of ĝ satisfying γ_{1}(x) + (∂ψ_{1}/∂x_{2})(x)γ_{2}(x) = γ̂_{1}(Ψ(x)), (∂ψ_{2}/∂x_{2})(x)γ_{2}(x) = γ̂_{2}(Ψ(x)), and γ̂_{i} = 0 whenever i > 2.

We will not attempt a detailed study of rank 2 involution systems, but we will look at a few special cases. There is no need to discuss the finite dimensional Lie algebra case, with σ and τ constant (and σ = 0 w.l.o.g.). Here (2.7) and (2.8) are applicable. In the following it will always be assumed that σ or τ is not constant. As was seen in sections 3 and 4, both σ and τ are first integrals of f, and application of L_{g} and L_{h} produces more first integrals. In fact, for this reason there are rank 2 involution systems which do not occur as systems of infinitesimal orbital symmetries of a nonzero vector field f. For example, let γ_{2} = exp (− (x_{1}x_{2} + x_{1}^{2}x_{3} + … + x_{1}^{n−1}x_{n} + x_{1}^{n+1})), and hence σ = x_{2} + 2x_{1}x_{3} + … + (n − 1)x_{1}^{n−2}x_{n} + (n + 1)x_{1}^{n}. Now σ and all ∂^{k}σ/∂x_{1}^{k} are first integrals of any vector field f admitting the infinitesimal orbital symmetries g and h, and this set contains n independent functions, forcing f = 0.

In any case, whenever the procedure outlined above produces at least two independent first integrals, the intuitive expectation that there should be a reduction of dimension by at least two is satisfied. Let us therefore consider the “borderline” case when this procedure yields exactly one first integral, up to functional dependence. We may assume that σ is not constant. If L_{g}(σ) = L_{h}(σ) = 0 then restricting to a level set of σ will lead to the finite dimensional Lie algebra case, with the dimension reduced by one. If L_{g}(σ) ≠ 0 then h̃ := L_{g}(σ)h−L_{h}(σ)g still normalizes f, and admits the first integral σ. Furthermore [g, h̃] = σ̃g + τ̃h̃ and σ̃ = 0 follows from L_{[g,h̃]}(σ) = Lh̃L_{g}(σ) − L_{g}Lh̃(σ) = 0. (Recall that L_{g}(σ) is locally a function of σ on an open-dense subset of U, and is therefore annihilated by Lh̃.) Therefore we may assume that [g, h] = τh, with τ not constant. (The case L_{g}(τ) ≠ 0 works similarly.)

Locally, we have h = (1, 0, …, 0)^{t}, and g = (γ_{1}, γ_{2}, 0, …, 0)^{t} after a suitable choice of coordinates, and ∂γ_{2}/∂x_{1} = 0, τ = −∂γ_{1}/∂x_{1} . Since γ_{2} is a function of x_{2}, …, x_{n} only, there is a function ψ_{2} of x_{2}, …, x_{n} such that (∂ψ_{2}/∂x_{2})γ_{2} = 1. Therefore, after applying the invertible transformation Ψ(x) = (x_{1}, ψ_{2}(x_{2}, …, x_{n}), x_{3}, …, x_{n})^{t}, there remains g = (γ_{1}, 1, 0, …, 0)^{t}.

Now L_{h}(τ) = ∂τ/∂x_{1} is locally a function of τ if and only if there are functions ρ (of n − 1 variables) and ν (of one variable) such that τ(x) = ν(x_{1} + ρ(x_{2}, …, x_{n})). (To see “if”, note that ν′ is locally a function of ν near any point z such that ν′(z) ≠ 0. The other implication follows from solving an elementary differential equation ∂τ/∂x_{1} = β(τ), with x_{2}, …, x_{n} as parameters.) Therefore, locally τ(x) = μ′(x_{1} + ρ(x_{2}, …, x_{n})), and γ_{1}(x) = −μ(x_{1} + ρ(x_{2}, …, x_{n})) + σ(x_{2}, …, x_{n}), with suitable functions μ and σ. With L_{g}(τ) = μ′′(x_{1} + ρ) (−μ(x_{1} + ρ) + σ + ∂ρ/∂x_{2}), and the observation that μ′′ and μ are locally functions of μ′, it follows that L_{g}(τ) is locally a function of τ (equivalently, of x_{1} + ρ) if and only if σ + ∂ρ/∂x_{2} is. But this is possible only if σ + ∂ρ/∂x_{2} = c = const., as differentiation with respect to x_{1} shows. We have found

h(x) = ( 1 0 0 ⋮ 0 ) , g(x) = ( −μ(x_{1} + ρ(x_{2}, …, x_{n})) − (∂ρ/∂x_{2})(x_{2}, …, x_{n}) + c ; 1 ; 0 ; ⋮ ; 0 ) ,

and one may assume c = 0.

Let us take a brief look at a vector field f centralized by both g and h: Such a vector field is independent of x_{1} (due to h), and admits the first integral ψ := x_{1} + ρ. Evaluating [g, f] = 0 shows that, in addition, the last n − 1 entries of f are independent of x_{2} . Thus,

f(x) = ( \begin{matrix}ϕ(x_{2}, …, x_{n}) \\ f̂(x_{3}, …, x_{n}) \end{matrix} ) ,

and this, together with the first integral condition, is necessary and sufficient. The solution of such an equation requires solving a differential equation in K^{n−2} for x_{3}, …, x_{n}, then a quadrature (for x_{2} ). Since ψ is a first integral, the first entry of a solution can be found from x_{1}+ρ(x_{2}, …, x_{n}) = const. So it may be said that the first integral saves one quadrature, compared to the usual procedure.

Perhaps a “concrete” example is in order. On K^{3} , consider

f̃(x) := ( \begin{matrix}x_{1} + x_{1}^{2}x_{3} \\ 2x_{1} + x_{2} + x_{1}x_{2}x_{3} − x_{1}^{3}x_{3}^{2} \\ −2x_{3} − 2x_{1}x_{3}^{2} \end{matrix} ) ,

g̃(x) := ( \begin{matrix}x_{1}^{3}x_{3} \\ x_{1}^{2}x_{2}x_{3} + x_{1} \\ −x_{1}^{2}x_{3}^{2} \end{matrix} ) , h̃(x) := ( \begin{matrix}x_{1} \\ x_{2} \\ −x_{3} \end{matrix} ) .

Then a simple verification shows that g̃ and h̃ centralize f̃, while [g̃, h̃] =

−x_{1}^{2}x_{3}h̃. (Admittedly, this example has been built in an artificial manner, but it should be noted that there is a manageable access to g̃ and h̃: One can restrict the search to centralizer elements that are polynomials of a prescribed degree, and determining these amounts to solving a finite system of linear equations. The unusual feature here is that one can find nontrivial polynomial vector fields centralizing f̃.)

The map Ψ(x) := (exp(x_{1}), x_{2} exp(x_{1}), x_{3} exp(−x_{1}))^{t} straightens h̃; more precisely, DΨ(x)h(x) = h̃(Ψ(x)), with h(x) = (1, 0, 0)^{t}. (Although there are many straightening maps, this one may be considered natural.) Furthermore, Ψ transforms g̃ to g, and f̃ to f, with

g(x) = ( \begin{matrix}x_{3} exp(x_{1}) \\ 1 \\ 0 \end{matrix} ) , and f(x) = ( \begin{matrix}1 + x_{3} \\ 2 − x_{3}^{2} \\ −x_{3} − x_{3}^{2} \end{matrix} ) ,

as a routine computation shows. The equation ẋ = f(x) can easily be solved by elementary functions, and applying Ψ produces the general solution of ẋ = f̃(x). (There also is the first integral x_{1}^{2}x_{3} for f̃.)

5.3. Can there be too much reduction? From the point of view of (2.2), this can happen: If the g_{i} admit no common first integrals then this proposition is not applicable, and it is also of little use when there is only one common first integral, up to functional dependence, in the orbitally symmetric case. (It can be easily achieved that, say, the first entry of f is independent of x_{2}, …, x_{n}: Just divide f by a suitable scalar function.) Let us see that things cannot be too bad even in this situation. (Hypotheses and notation are as in §2, except that there is no restriction on s.) Assume that there are no first integrals to be gained from the strategies discussed in §4, and that we deal with the nondegenerate case of §4, with the g_{i} centralizing f; compare (4.4) to see that there is no loss of generality in these assumptions. Then the g_{i} span a finite dimensional Lie algebra L of vector fields, of dimension r < n, and the local group orbits generically are rdimensional. If r = n − 1, and thus there are no useful common first integrals of the g_{i}, structure theory of finite dimensional Lie algebras is helpful: The radical is a solvable ideal, and thus yields a preliminary reduction by (2.8), which requires only quadratures and finding inverses of transformations. By (2.6) and (2.7), the quotient modulo the radical still survives as algebra of infinitesimal symmetries for the reduced vector field. In the semisimple case, one can always resort to (maximal) toral subalgebras, or to solvable subalgebras, and obtain at least a partial reduction. There seem to be no serious problems in such situations.

Acknowledgement. I am grateful to Peter J. Olver for valuable comments on an earlier version of this paper. Also, I thank the referees for their careful reading of the manuscript and their suggestions.

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Zentrum Mathematik TU München D- 80290 München, Germany walcher@mathematik.tu-muenchen.de

Received March 10, 1998 and in final form August 4, 1998