  This work considers two related families of nonlinear and nonlocal problems
in the plane $\R^2$. The first main result derives the general integrable
solution to a generalized Liouville equation using the Wronskian of two coprime
complex polynomials. The second main result concerns an application to a
generalized Ladyzhenskaya--Gagliardo--Nirenberg interpolation inequality, with
a single real parameter $\beta$ interpreted as the strength of a magnetic
self-interaction. The optimal constant of the inequality and the corresponding
minimizers of the quotient are studied and it is proved that for $\beta \ge 2$,
for which the constant equals $2\pi\beta$, such minimizers only exist at
quantized $\beta \in 2\N$ corresponding to nonlinear generalizations of Landau
levels with densities solving the generalized Liouville equation. This latter
problem originates from the study of self-dual vortex solitons in the abelian
Chern--Simons--Higgs theory and from the average-field-Pauli effective theory
of anyons, i.e.\ quantum particles with statistics intermediate to bosons and
fermions. An immediate application is given to Keller--Lieb--Thirring stability
bounds for a gas of such anyons which self-interact magnetically (vector
nonlocal repulsion) as well as electrostatically (scalar local/point
attraction), thus generalizing the stability theory of the 2D cubic nonlinear
Schr\"odinger equation.
