Classify translating configurations for Λ=2 in the plane

Determine a complete classification of translating equilibria of point vortices with strengths −1 and 2 (circulation ratio −2) in two-dimensional inviscid flow, i.e., all solutions of k + ∑_{j≠i} Q_j/(z_i − z_j) = 0 with Q_i ∈ {−1, 2}, including their parameter dependence and structural properties, given that existing factorization and τ-function methods do not apply when Λ=2.

Background

Throughout the paper, translating equilibria are characterized by the algebraic system k + ∑_{j≠i} Q_j/(z_i − z_j) = 0. For Λ=1 (two species with strengths ±1), all translating configurations can be generated via the Baker–Akhieser function and Darboux transformations for the Schrödinger operator, and their complete classification is known. In contrast, for Λ=2 (two species with strengths −1 and 2) the associated Lax operator is third order (Sawada–Kotera), and the Baker–Akhieser framework does not reduce to the Schrödinger form needed to recover the bilinear translating equilibrium equation.

Because of this structural obstacle, the methods that yield a full description in the Λ=1 case do not extend to Λ=2. The authors explicitly note that the classification of translating configurations for Λ=2 remains unresolved.