In the case when $M$ is equipped with a bi-Hamiltonian structure $(M,\pi_1, \pi_2)$ we show how to construct family of Poisson structures on the tangent bundle $TM$ to a Poisson manifold. Moreover we present how to find Casimir functions for those structures and we discuss some particular examples.
"Bi-Hamiltonian Structures on the Tangent Bundle to a Poisson Manifold." J. Geom. Symmetry Phys. 52 47 - 66, 2019. https://doi.org/10.7546/jgsp-52-2019-47-66