Explicit Conservation Laws for the Normal Forms of Two-Dimensional gl-Regular Nijenhuis Operators

Bolsinov, Konyaev and Matveev asked in 2024 to describe all conservation laws of the normal forms of two-dimensional \(\mathrm{gl}\)-regular Nijenhuis operators, or at least to find one regular conservation law for each of them. A conservation law of \(L\) is a function \(F\) with \(\mathrm{d}(L^*\mathrm{d} F)=0\), regular if \(\mathrm{d} F\) and \(L^*\mathrm{d} F\) are linearly independent; everything is local and real analytic. The existence of a regular law was known, by the Cauchy–Kovalevskaya theorem, as were explicit laws at algebraically generic points and the laws \(\operatorname{tr} F(L)\) at differentially non-degenerate points. For every normal form and all values of its parameters we give an explicit regular conservation law. For the singular series \(O\), \(P\) and \(S\) it has the form \(f_1(\lambda_1)+f_2(\lambda_2)\) anticipated in the problem, with \(f_i\) a finite sum of negative fractional powers, or a logarithm, of the eigenvalue; the lower-order coefficients solve a triangular linear system with unit pivots, and for \(k=1\) there are closed forms, for instance the analytic root \(\Phi\) of \(8y\Phi^3+6\Phi=c_0+3x\) for \(S^3_c\). We also describe all real-analytic conservation laws of every normal form; for \(O\), \(P\) and \(S\) they consist of an explicit finite principal part subject to finitely many linear conditions, at most one logarithm, and arbitrary analytic functions of explicit roots of the eigenvalues. Completeness is proved by complexification, Cauchy estimates on the fibres of the eigenvalue map and a monodromy argument. No printed answer was found in the sources we searched; one 2025 survey could not be consulted. This is an unrefereed note.