Symmetry algebras for superintegrable systems

A class of elliptic solutions to the matrix CKP equation is studied. Equations of motion for their poles and matrix coefficients at the poles are obtained. As in the scalar case, they are of the first order, in contrast to what takes place in the KP and BKP hierarchies, where the equations of motion are of the second order.

nlin.SI↗

arXiv2026

On integrals of non-autonomous dynamical systems in finite characteristic

We use a difference Lax form to construct simultaneous integrals of motion of the fourth Painlevé equation and the difference second Painlevé equation over fields with finite characteristic $p>0$. For $p\neq 3$, we show that the integrals can be normalised to be completely invariant under the corresponding extended affine Weyl group action. We show that components of reducible fibres of integrals correspond to reductions to Riccatti equations. We further describe a method to construct non-rational algebraic solutions in a given positive characteristic. We also discuss a projective reduction of the integrals.

nlin.SI↗

arXiv2026

On equivalence between solution structures of the anti-self-dual Yang--Mills equations

We establish an exact correspondence between the quasi-Wronskian and quasi-Grammian constructions for the anti-self-dual Yang--Mills (ASDYM) equations with complex $2\times2$ matrix-valued fields. Starting from vacuum wave functions associated with $2M$ pairwise distinct nonzero constant spectral parameters, we restrict attention to a nonempty local domain on which the corresponding interpolation matrix is invertible. We show that both constructions are governed by the same uniquely solvable matrix-polynomial interpolation problem. In particular, the quasi-Wronskian Darboux polynomial $T(λ)$ and the quasi-Grammian rational dressing matrix $W(λ)$ satisfy \[ T(λ)=q_ξ(λ)W(λ), \] where $q_ξ(λ)$ is the monic scalar polynomial whose zeros are the prescribed poles of $W(λ)$. This identity yields explicit relations between the corresponding Yang and potential matrices and shows that the two constructions generate identical ASDYM gauge potentials in the same gauge.

nlin.SI↗