Geometric models for Lie--Hamilton systems on ${\mathbb{R}}^2$

This paper provides a geometric description for Lie–Hamilton systems on ${\mathbb{R}}^2$ with locally transitive Vessiot–Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie–Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie–Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give rise to a natural framework for the analysis of Lie–Hamilton systems on ${\mathbb{R}}^2$ while retrieving known results in a natural manner. Our methods may be extended to study Lie–Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.