Negative powers of Olver recursion operators

Olver’s concept of a recursion operator for symmetries of an evolution equation is extended to include negative powers of the operator. Some negative order KdV equations are derived in this way. In particular, the inverse image of the trivial zero symmetry generator has an elegant formulation in terms of an eigenfunction of the associated Schrödinger equation used in the inverse scattering solution of the KdV equation and this formulation is used to show that this new equation is the Miura transform of a sinh-Gordon equation.