  A large class of new exact solutions to the steady, incompressible Euler
equation on the plane is presented. These hybrid solutions consist of a set of
stationary point vortices embedded in a background sea of Liouville-type
vorticity that is exponentially related to the stream function. The input to
the construction is a "pure" point vortex equilibrium in a background
irrotational flow. Pure point vortex equilibria also appear as a parameter $A$
in the hybrid solutions approaches the limits $A\to 0,\infty$. While $A\to 0$
reproduces the input equilibrium, $A\to\infty$ produces a new pure point vortex
equilibrium. We refer to the family of hybrid equilibria continuously
parametrised by $A$ as a "Liouville link". In some cases, the emergent point
vortex equilibrium as $A\to\infty$ can itself be the input for a second family
of hybrid equilibria linking, in a limit, to yet another pure point vortex
equilibrium. In this way, Liouville links together form a "Liouville chain". We
discuss several examples of Liouville chains and demonstrate that they can have
a finite or an infinite number of links. We show here that the class of hybrid
solutions found by Crowdy (2003) and by Krishnamurthy et al. (2019) form the
first two links in one such infinite chain. We also show that the stationary
point vortex equilibria recently studied by Krishnamurthy et al. (2020) can be
interpreted as the limits of a Liouville link. Our results point to a rich
theoretical structure underlying this class of equilibria of the 2D Euler
equation.
