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This work considers two related families of nonlinear and nonlocal problems in the plane 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 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 2, for which the constant equals 2, such minimizers only exist at quantized 2 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.
Ataei et al. (Sun,) studied this question.