Classification of the Calogero equations with respect to point transformations

Abstract

The article is devoted to the problem of classification of Calogero-type equations with respect to point transformations. It is known that Calogero equations can be reduced to linear equations using contact transformations, provided that the function f is quadratic. However, for an arbitrary function f the equivalence problem is open. Admissible point transformations that preserve the class of Calogero equations are considered. We construct differential invariants with respect to such transformations and apply them to solve the equivalence problem.