Symmetries of Integro-Differential Equations: A Survey of Methods Illustrated by the Benney Equations

Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as in the case of infinite systems of differential equations. This paper is aimed to survey the modern approaches to symmetries of integro-differential equations. As an illustration, an infinite symmetry Lie algebra is calculated for a system of integro-differential equations, namely the well-known Benney equations. The crucial idea is to look for symmetry generators in the form of canonical Lie-Backlund operators.