A Necessary Condition for Existence of Lie Symmetries in General Systems of Ordinary Differential Equations

Abstract

Lie symmetries for ordinary differential equations are studied. In systems of ordinary differential equations, there do not always exist non-trivial Lie symmetries around equilibrium points. We present a necessary condition for existence of Lie symmetries analytic in the neighbourhood of an equilibrium point. In addition, this result can be applied to a necessary condition for existence of a Lie symmetry in quasihomogeneous systems of ordinary differential equations. With the help of our main theorem, it is proved that several systems do not possess any analytic Lie symmetries.

The complete symmetry group of the generalised hyperladder problem

“…They were particularly interested in the symmetries Y n m , m = 1, n − 1 which have polynomial coefficient functions and hence have the property of being analytic symmetries. This was a feature of relevance to their earlier work [5]. This particular set of symmetries possesses an Abelian algebra and from this property Imai and Hirata were able to conclude that the ladder system was integrable.…”

“…In two recent papers Imai and Hirata developed a necessary condition for the existence of Lie point symmetries in n-dimensional systems of first-order ordinary differential equations [5] and applied the ideas developed there to establish a new integrable family in the class of Lotka-Volterra systems [6]. Of the infinite number of Lie symmetries that such a system possesses Imai and Hirata [5] were concerned with autonomous symmetries of the form…”

“…Of the infinite number of Lie symmetries that such a system possesses Imai and Hirata [5] were concerned with autonomous symmetries of the form…”

“…In their application of the results of [5] to n-dimensional homogeneous, that is to say quadratic, Lotka-Volterra systems of the forṁ…”

“…The ladder system has certain general properties [3,5,6]. According to [3] the rank of the matrix A is two for n > 2 and the n-dimensional ladder system can be considered as the simplest Riccati equation for n i=1 x i and as such is an example of a decomposed system [4].…”