Quadratic Hamiltonians on non-symmetric Poisson structures

Many dynamical systems may be represented in a set of non-canonical coordinates that generate an su(2) algebraic structure. The topology of the phase space is the one of the S2 sphere, the Poisson structure is the one of the rigid body, and the Hamiltonian is a parametric quadratic form in these “spherical” coordinates. However, there are other problems in which the Poisson structure losses its symmetry. In this paper we analyze this case and, we show how the loss of the spherical symmetry affects the phase flow and parametric bifurcations for the bi-parametric cases.