  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.
