  It is well known that functions in involution with respect to Poisson
brackets have a privileged role in the theory of completely integrable systems.
Finding functionally independent functions in involution with a given function
$h$ on a Poisson manifold is a fundamental problem of this theory and is very
useful for the explicit integration of the equations of motion defined by $h$.
In this paper, we present our results on the study of the inverse, so to speak,
problem. By developing a technique analogous to that presented in P. Damianou
and F. Petalidou, Poisson brackets with prescribed Casimirs, Canad. J. Math.,
2012, vol. 64, 991-1018, for the establishment of Poisson brackets with
prescribed Casimir invariants, we construct an algorithm which yields Poisson
brackets having a given family of functions in involution. Our approach allows
us to deal with bi-Hamiltonian structures constructively and therefore allows
us to also deal with the completely integrable systems that arise in such a
framework.
