## Abstract

Computing the Lie point symmetries of *systems* of linear differential equations can be prohibitively difficult. For homogeneous systems in Kovalevskaya form of order two or higher, this paper proves the existence of a basis of infinitesimal generators (as determined by Lie's algorithm) whose characteristic forms are homogeneous in the dependent variables of degree zero, one or two. Suppose Lie's algorithm yields a characteristic form of degree two; in this case, the system is second order. If it contains only ordinary differential equations (ODEs), its general solution is constructed from those of a given *first-order* linear homogeneous system, of the same dimension, and a second-order linear *scalar* equation. Otherwise, coordinates are given in which its dimension (necessarily two or higher now) is lowered by one, leaving an inhomogeneous system of parametrized ODEs. Its homogeneous part is solved as in the previous case.

## 1. Introduction

Methods based on Lie groups of point symmetries are indispensable for the systematic analysis, simplification and solution of differential equations [1–4]. Lie's algorithm for calculating these groups is well known: the (otherwise intractable) symmetry condition is linearized to obtain an overdetermined system of linear homogeneous ‘determining equations’ for the group generators. However, the size and complexity of this system can make solving it impractical. (Consider that Maple 17's ‘Infinitesimals’ command, executed without ‘Hints’ on a contemporary high-end personal computer, can take over 4 h to determine the generators admitted by the system of three second-order linear ordinary differential equations (ODEs) in §5*c*.) This is the main reason why constraints on the Lie point symmetries of classes of differential equation are so useful; for example, see [1,2,4–8]. In this paper, labour-saving constraints on the Lie point symmetry generators admitted by systems of linear differential equations are derived.

Prior knowledge of the form of Lie point symmetries can also lead to new symmetry methods. On page 476 of [9], Lie proves that if a (1+1)-dimensional second-order linear scalar partial differential equation (PDE) cannot be integrated using Monge's method,^{1} its Lie point symmetries are affine bundle maps. In other words, they are projectable/fibre-preserving point transformations that are affine in the dependent variable at each point. Lie, whose proof actually applies to all contact symmetries, uses the result to integrate these equations systematically.

In §27 of [10], Ovsiannikov proves that for all the second-order linear scalar PDEs, Lie's algorithm determines generators of affine bundle map Lie point symmetries only, with one requirement (see page 183 of [10]): they cannot ‘degenerate’ in any set of independent variables into ODEs. Their Lie point symmetries are of course not all affine bundle maps; see, for example, §3.3 of [3]. This result is finally extended, without the need for additional hypotheses, to all higher order linear scalar ODEs and PDEs by Bluman in Theorems VI and VIII of [11] (wherein the Lie point symmetry generators admitted by certain classes of quasi-linear scalar equations are also partially determined). As for linear *systems*, there does not appear to be an analogous result in the literature.

This paper applies to systems of linear ODEs and PDEs, in Kovalevskaya form and of order two or higher. Its purpose is to clarify the functional forms of the Lie point symmetry generators determined by Lie's algorithm. In particular, if Lie's algorithm determines a generator of non-affine bundle map Lie point symmetries, the system's general solution is constructed from the solutions to a given first-order system and scalar equation. The main conclusions are summarized in §2, with proofs and examples given in §§3–5, respectively. Some related results are discussed in §6.

## 2. Preliminaries and main results

Let **x**=(*x*^{1},…,*x*^{N}) and **u**=(*u*^{1},…,*u*^{M}) denote the independent and dependent variables, respectively, in the *n*th-order (*n*>1) system
*M* and repeated Latin indices from 1 to *N*. Furthermore, unless otherwise stated, expressions involving free indices hold for all Greek indices ranging from 1 to *M* and all Latin indices from 1 to *N*.

All (unordered) multi-indexes **J**=(*j*_{1},…,*j*_{R}), 1≤*j*_{i}≤*N*, in (2.1) are summed over from those of length 1 to *n*. The total derivative operators *D*_{J} are defined in the usual way
*A*^{αJ}_{β} are symmetric in the components of **J** and *A*^{α1}_{β}=0 where **1** is the *n*-vector (1,…,1).

Each one-parameter (local) Lie group of point symmetries of (2.1) has the form
*X*=*ξ*^{k}∂_{xk}+*η*^{α}∂_{uα} are determined by solving the linearized symmetry condition (LSC) [4],
**u** vanish separately; these are the ‘determining equations’ for *X*. Note that without an additional *local solvability* hypothesis, the LSC is a sufficient but not necessary condition for *X* to generate the symmetry group (2.2). In particular, (2.1) is locally solvable if (but not only if) *A*^{αJ}_{β} and

The main results proved in this paper are summarized in the following two theorems.

### Theorem 2.1

*If system* (*2.1*) *is third order or higher, LSC* (*2.3*) *determines generators of only affine bundle maps. Moreover, they are linear combinations of generators of the form
**If* (*2.1*) *is second order, the generators determined by LSC* (*2.3*) *are linear combinations of generators of form* (*2.4*) *and those of the form*

### Theorem 2.2

*Suppose LSC* (*2.3*) *determines a non-zero generator of the form* (*2.5*), *in which case n*=2. *The dependent variables can be relabelled (if necessary) so that* *. Once this is done, in the canonical coordinates defined by* *the transformation* *effectively sets* *and system* (*2.1*) *can still be solved for* *. Assume these changes have been made retrospectively, so that* *and L*^{α}[**u**]=0 *is still solved for*

*Now, any derivatives of u*^{2}*,…,u*^{M} *in* (*2.1*) *are taken with respect to x*^{1} *only, while* (*2.5*) *is*
*If u*^{1} *is differentiated with respect to x*^{1} *only, the general solution to (2.1) is* *, where* *is a* 2×*M matrix of arbitrary constants,* *satisfies*
*and R*^{1}(**x**), *R*^{2}(**x**) *are any two linearly independent solutions to*
*Otherwise, M>1 and for arbitrary functions f(x*^{2}*,…,x*^{N}*), g(x*^{2}*,…,x*^{N}*),
**Substituting* (*2.9*) *into* (*2.1*) *yields an inhomogeneous system of ODEs for u*^{2}*,…,u*^{M} *parametrized by x*^{2}*,…,x*^{N}*. Let* (*2.1*) *represent its homogeneous part* (*with the dependent variables relabelled appropriately and M one less than its original value*). *Its general solution is also* *, as defined previously.*

## 3. Systems of order three or higher

The following lemma is immediate.

### Lemma 3.1

*Let* *n*≥3. *Eliminating* *from LSC* (*2.3*) *using system* (*2.1*) *does not alter terms of the form* *Furthermore, if* *it does not alter terms of the form*

### Lemma 3.2

*Let* *n*≥3. *LSC* (*2.3*) *determines generators of affine bundle maps only*.

### Proof.

Let *n*≥3. Before eliminating **u** have the form **u** have the form *n* increases, yields
*n*=3 and λ=*n*(*n*−1)/2 for *n*>3. Therefore, by lemma 3.1 the *k*>1. They come from

Finally, apply the argument again to the

### Corollary 3.3

*Let* *n*≥3. *The generators determined by LSC* (*2.3*) *are linear combinations of generators of form* (*2.4*).

## 4. Systems of order two

If *n*=2, system (2.1) has the form

### Lemma 4.1

*Let* *n*=2. *Eliminating* *from* (*4.2*) *using* (*4.1*) *does not alter terms of the form* *nor terms involving* *η*^{μ} *in which the derivatives of* **u** *have the form*

### Lemma 4.2

*Let* *n*=2. *The generators determined by* (*4.2*) *are linear combinations of generators of form* (*2.4*) *and* (*2.5*).

### Proof.

Let *n*=2. Before eliminating **u** have the form **u** have the form *ξ*^{1} is linear in **u**. Similarly, by lemma 4.1, before and after eliminating *k*>1, terms are now

Before eliminating *η*^{μ} and derivatives of **u** of the form *η*^{μ} do involve *u*^{α}∂_{uα} completes the argument. ▪

### Corollary 4.3

*Suppose LSC* (*4.2*) *determines a non-zero generator of the form* (*2.5*), *in which case* *n*=2. *If* *L*^{α}[**h**(**x**)]=0 *then there exist systems of second- and third-order linear PDEs* *respectively* (*symmetric in* *α*, *β* *and* *γ*) *such that*
*here only the highest order derivatives of* **h**(**x**) *are shown*.

### Proof.

Let the premise of the corollary hold and denote (2.5) by *X*. If *L*^{α}[**h**(**x**)]=0, then LSC (4.2) also determines the generator
**h** in each infinitesimal are displayed. By lemma 4.2, this is identically zero—the *u*^{α}*u*^{β}∂_{xk} and *u*^{α}*u*^{β}*u*^{γ}∂_{uλ} coefficients are

### Lemma 4.4

*Suppose LSC* (*4.2*) *determines a non-zero generator of form* (*2.5*)*, in which case* *n*=2. *The dependent variables can be relabelled* (*if necessary*) *so that* *Once this is done, in the canonical coordinates defined by* *the transformation* *effectively sets* *and system* (*4.1*) *can still be solved for* *. Assume these changes have been made retrospectively, so that* *and* *L*^{α}[**u**]=0 *is still solved for* *Now* *for* *p*>1 *and*

### Proof.

Let the premise of the lemma hold. Suppose first that

Invoking corollary (4.3), and using the fact that (4.1) is in Kovalevskaya form, there exist functions

It follows directly that in coordinates defined by

The rest of the theorem follows by eliminating **h**. ▪

Theorem 2.2 can be proved now:

### Proof.

Suppose LSC (4.2) determines a non-zero generator of form (2.5), in which case *n*=2. Invoking lemma 4.4, use coordinates in which (4.1) when *α*=1 is an ODE and *q*>1, coefficients at *p*=1 then *p*>1 yields
*u*^{β}*u*^{γ} coefficients, *β*>1, impose no further constraint.) Define **R**(**x**) and **C** as in theorem 2.2. According to (4.7), *α*=1,…,*M*. Therefore, the general solution **u** to (4.1) satisfies **Q**^{−1}**u**=**C****R**.

Alternatively, now suppose (without loss of generality) that there exists an index *α*>1 such that *β*=1 then *β*>1 yields *β*>1. The *p*,*β*>1 are *p*,*β*>1, and so by (4.5) only *u*^{1} is differentiated with respect to variables other than *x*^{1}.

For *α*=*β*>1, the *β*>1 at *α*=1 are *β*>1. Therefore, (2.9) holds for arbitrary functions *f*(*x*^{2},…,*x*^{N}) and *g*(*x*^{2},…,*x*^{N}). Inserting (2.9) into (4.1) yields an inhomogeneous system of ODEs for *u*^{2},…,*u*^{M} parametrized by *x*^{2},…,*x*^{N}.

For *β*>1, the *M* one less than its original value), its general solution **u** also satisfies **Q**^{−1}**u**=**C****R**, with **Q**, **R** and **C** defined as in theorem 2.2. ▪

### Remark 4.5

Suppose the LSC yields a non-zero generator of form (2.5). Furthermore, suppose that in the coordinates derived in lemma 4.4 (i.e. those in which *L*^{α}[**u**]=0 contains PDEs. For the sake of completeness, the constraints this imposes on *L*^{α}[**u**]=0 are as follows.

In this case, *n*=2 and *L*^{α}[**u**]=0 is given by (4.1). Invoking lemma 4.4, change retrospectively to coordinates in which (4.1) when *α*=1 is an ODE,

In LSC (4.4), the *q*>1 coefficients are (4.5) and the *α*>1 such that *u*^{β}*u*^{γ}, respectively) impose no further constraint.

## 5. Examples

The following examples illustrate some of the consequences of invariance under non-affine bundle map Lie point symmetries.

### (a) The two-dimensional systems of ordinary differential equations with non-affine bundle map Lie point symmetries

Let *M*=2, *N*=1 and suppose the LSC determines a non-zero generator of form (2.5), in which case *n*=2. Locally, coordinates exists in which *L*^{α}[**u**]=0, *α*=1,2, still has form (4.1); use these coordinates retrospectively.

For simplicity, denote *x*^{1}=*x*, *u*^{1}=*u* and *u*^{2}=*v*. The only constraints on (4.1) are (4.7) and (4.8). According to (4.8), the system has the form, in terms of an arbitrary constant *c*_{1},
**u**:
*α*=1,2 and *α*,*β*=1,2. Therefore, *c*_{i}, this determines four families of ODEs:
*M*^{2}+4*M*+3) of linearly independent Lie point symmetry generators; see González-Gascon & González-Lopez, [12]. Therefore, they can be transformed using (local) point transformations into the free particle system *u*_{,xx}=*v*_{,xx}=0 [13]. Moreover, the first system is a *contraction* of the second [14], as can be seen by taking the limit as

For the first, second, third and fourth family, respectively, the generator (2.6) is

### (b) Some two-dimensional systems of constant-coefficient partial differential equations with non-affine bundle map Lie point symmetries

Let *M*=2 and suppose the LSC determines a non-zero generator of form (2.5), in which case *n*=2. Locally, coordinates exists in which *L*^{α}[**u**]=0, *α*=1,2, has the form
*β*=1,2 and for *p*,*q*>1,
*c*_{i}, and denoting *u*^{1}=*u*, *u*^{2}=*v*, the solutions to this system determine the family of PDEs

### (c) Solving a three-dimensional system of ordinary differential equations using its non-affine bundle map Lie point symmetries

Let *M*=3 and denote *x*^{1}=*x*, *u*^{1}=*u*, *u*^{2}=*v* and *u*^{3}=*w*. The LSC for the system
*x*^{2}/2, and a 2×3 matrix *M*^{2}+4*M*+3) linearly independent Lie point symmetry generators, so it cannot be transformed into the free particle system *u*_{,xx}=*v*_{,xx}=*w*_{,xx}=0 [13].

### (d) Solving a two-dimensional system of partial differential equations using its non-affine bundle map Lie point symmetries

Denote *x*^{1}=*x*, *x*^{2}=*t*, *u*^{1}=*u* and *u*^{2}=*v*. The LSC for the system of PDEs
*x*,*t*,*u*,*v*)↦(*x*,*t*−*x*,*u*+e^{2x}*v*,*v*) sets *f*_{1}(*t*) and *f*_{2}(*t*), (2.9) becomes *u*=*f*_{1}+*f*_{2}e^{x} and so
*v* then reverting back to the original coordinates yields the solution to the original system: for arbitrary functions *g*_{i}=*g*_{i}(*t*−*x*) and with *g*_{1}′ denoting the derivative of *g*_{1},

## 6. Discussion

For *systems* of linear ODEs and PDEs, of order two or higher and in Kovalevskaya form, this paper establishes labour-saving constraints on the generators determined by Lie's algorithm. Using any given generator of non-affine bundle map Lie point symmetries (as determined by Lie's algorithm), the general solution was constructed from the solutions to a given first-order system and scalar equation. For any number of dependent variables, examples of linear systems with these generators can be found easily by solving (4.7) and (4.8), or the constraints listed in remark 4.5. Theorem 2.2 implies a practical way to avoid such systems: ensure that a linear scalar second-order ODE cannot be extracted from (2.1). This is a necessary but not sufficient condition for Lie's algorithm to return generators of non-affine bundle map Lie point symmetries, as the system *u*_{,xx}=*u*, *v*_{,xx}=0 shows.

If infinitesimal generators are made easier to compute, so are the corresponding group-invariant solutions. Broadbridge & Arrigo show in [15] that by incorporating these solutions within generators admitted by linear scalar equations, additional ‘generations’ of group-invariant solutions can be obtained. In the light of theorems 2.1 and 2.2, the proof of this fact (theorem 1 in [15]) applies, with only trivial modifications, to the systems addressed herein.

Similar results to those discussed in this paper have already been proved for large classes of nonlinear scalar equations. For example, theorem 4.11 in Heredero & Olver, [16] states that the *spatial* Lie point symmetries, i.e. those that do not alter time, of evolutionary-type PDEs are affine bundle maps. These are PDEs in which a derivative comprising at least one time derivative is equated to a differential function of only spatial derivatives. In §6 of [16], this result is generalized so the differential function may also depend on time derivatives, although only in the special case of one spatial variable.

This paper focuses on Lie groups of point transformations, but the symmetry analysis of linear systems can be conducted in the more general framework of symmetry operators (SOs): linear differential operators that map solutions to solutions (this definition can be relaxed to include nonlinear integro-differential operators [17]). The method of separation of variables for the second-order linear PDEs is developed systematically using SOs by Miller [18]; an extension of this method (using sets of *non*commuting operators) is described by Shapovalov & Shirakov [19]. Fushchich & Nikitin [17] use SOs to investigate the symmetries of the equations of quantum physics and field theory. Therein, those that are not both linear and first-order are referred to as non-Lie symmetries.

In §5.2 of [4], Olver discusses the relationship between SOs and generalized symmetries of linear systems in the context of recursion operators. These are linear operators that act on generalized symmetries in characteristic form, returning possibly new characteristic forms. In particular, generators of Lie groups of affine bundle map point symmetries are in one-to-one correspondence with the first-order SOs.

Shapovalov & Shirakov establish in theorem 4.4 on page 702 of [20] a constraint on SOs that complements theorem 2.1: the second-order linear scalar PDEs (with three or more independent variables if they are not parabolic) do not admit nonlinear SOs. On a slightly different note, the same theorem states that up to the symmetries of linear superposition, the SOs of any given order admitted by these PDEs span a finite-dimensional vector space. In particular, therefore, up to the generators of linear superpositions they have finite-dimensional Lie point symmetry algebras. Other remarkable bounds on the dimensions and structures of the Lie symmetry algebras of linear ODEs are derived by Aguirre & Krause [21], Boyko *et al.* [22], Campoamor-Stursberg [23,24], González-Lopez [25], González-Gascon & González-Lopez [12] and Samokhin [26].

Working along slightly different lines, Kingston & Sophocleous describe [27] a powerful method for reducing PDEs to canonical forms, and for obtaining their point symmetries. Overcoming significant computational problems, they calculate the action of an arbitrary point transformation on the partial derivatives of scalar functions with two independent variables. They use these expressions to determine the nature of the ‘form-preserving’ point transformations that map certain classes of (1+1)-dimensional *n*th-order scalar PDEs to themselves (see, in particular, §4 of [27]).

Popovych *et al.* [28] use this approach to describe comprehensively the conservation laws and potential symmetries of (1+1)-dimensional linear scalar second-order parabolic PDEs. To this end, form-preserving transformations are used in the more formal context of ‘admissible’ transformations; classes of PDE are designated as ‘normalized’, ‘semi-normalized’ and ‘strongly semi-normalized’ based on the relationships between their form-preserving transformations, equivalence transformations and point symmetries. ‘Note 1’ on page 5 of [28] describes some of these relationships in the context of linear systems of PDEs, under the assumption that their equivalence groups consist of affine bundle maps.

## Acknowledgements

Many thanks to Prof. Peter Hydon for inspiring me to work on this problem, to Dr Linyu Peng for helpful comments on an early version of these results and to the anonymous referees for their constructive feedback.

## Footnotes

↵1 That is, if it does not possess an intermediate integral with one arbitrary function.

- Received November 22, 2013.
- Accepted February 27, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.