  A Lie-Hamilton system is a nonautonomous system of first-order ordinary
differential equations describing the integral curves of a $t$-dependent vector
field taking values in a finite-dimensional real Lie algebra of Hamiltonian
vector fields with respect to a Poisson structure. We provide new
algebraic/geometric techniques to easily determine the properties of such Lie
algebras on the plane, e.g., their associated Poisson bivectors. We study new
and known Lie-Hamilton systems on $\mathbb{R}^2$ with physical, biological and
mathematical applications. New results cover Cayley-Klein Riccati equations,
the here defined planar diffusion Riccati systems, complex Bernoulli
differential equations and projective Schr\"odinger equations. Constants of
motion for planar Lie-Hamilton systems are explicitly obtained which, in turn,
allow us to derive superposition rules through a coalgebra approach.
