Abstract. We study natural lifting operations from a bundle τ : E → R to the bundle π : J 1 τ * → E which is the dual of the first-jet bundle J 1 τ . The main purpose is to define a complete lift of a type (1, 1) tensor field on E and to understand all features of its construction. Various other lifting operations of tensorial objects on E are needed for that purpose. We prove that the complete lift of a type (1, 1) tensor with vanishing Nijenhuis torsion gives rise to a Poisson-Nijenhuis structure on J 1 τ * , and discuss in detail how the construction of associated Darboux-Nijenhuis coordinates can be carried out.

Abstract. This is a continuation of the work initiated in [18] on so-called driven cofactor systems, which are partially decoupling second-order differential equations of a special kind. The main purpose in [18] was to obtain an intrinsic, geometrical characterization of such systems, and to explain the basic underlying concepts in a brief note. In the present paper we address the more intricate part of the theory. It involves in the first place understanding all details of an algorithmic construction of quadratic integrals and their involutivity. It secondly requires explaining the subtle way in which suitably constructed canonical transformations reduce the HamiltonJacobi problem of the (a priori time-dependent) driven part of the system into that of an equivalent autonomous system of Stäckel type.

Abstract. Starting from a bundle τ : E → R, the bundle π : J 1 τ * → E, which is the dual of the first jet bundle J 1 τ and a sub-bundle of T * E, is the appropriate manifold for the geometric description of time-dependent Hamiltonian systems. Based on previous work, we recall properties of the complete lifts of a type (1, 1) tensor R on E to both T * E and J 1 τ * . We discuss how an interplay between both lifted tensors leads to the identification of related distributions on both manifolds. The integrability of these distributions, a coordinate free condition, is shown to produce exactly Forbat's conditions for separability of the time-dependent Hamilton-Jacobi equation in appropriate coordinates.