A class of recursion operators on a tangent bundle

We generalize the construction of a class of type (1, 1) tensor fields R on a tangent bundle which was introduced in a preceding paper. The generalization comes from the fact that, apart from a given Lagrangian, the further data consist of a type (1, 1) tensor J along the tangent bundle projection tau : TQ -> Q, rather than a tensor on Q. The main features under investigation are two kinds of recursion properties of R, namely its potential invariance under the flow of the given dynamics and the property of having vanishing Nijenhuis torsion. The theory is applied, in particular, to the case of second-order dynamics coming from a Finsler metric.