  We propose a method to construct first integrals of a dynamical system,
starting with a given set of independent infinitesimal symmetries. In the case
of two infinitesimal symmetries, a rank two Poisson structure on the ambient
space it is found, such that the vector field that generates the dynamical
system, becomes a Poisson vector field. Moreover, the symplectic leaves and the
Casimir functions of the associated Poisson manifold are characterized.
Explicit conditions that guarantee Hamilton-Poisson realizations of the
dynamical system are also given.
