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A large class of new exact solutions to the steady, incompressible Eulerequation on the plane is presented. These hybrid solutions consist of a set ofstationary point vortices embedded in a background sea of Liouville-typevorticity that is exponentially related to the stream function. The input tothe construction is a "pure" point vortex equilibrium in a backgroundirrotational 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 vortexequilibrium. We refer to the family of hybrid equilibria continuouslyparametrised by $A$ as a "Liouville link". In some cases, the emergent pointvortex equilibrium as $A\to\infty$ can itself be the input for a second familyof hybrid equilibria linking, in a limit, to yet another pure point vortexequilibrium. In this way, Liouville links together form a "Liouville chain". Wediscuss several examples of Liouville chains and demonstrate that they can havea finite or an infinite number of links. We show here that the class of hybridsolutions found by Crowdy (2003) and by Krishnamurthy et al. (2019) form thefirst two links in one such infinite chain. We also show that the stationarypoint vortex equilibria recently studied by Krishnamurthy et al. (2020) can beinterpreted as the limits of a Liouville link. Our results point to a richtheoretical structure underlying this class of equilibria of the 2D Eulerequation.

Liouville chains: new hybrid vortex equilibria of the 2D Euler equation