Two-dimensional Krall-Sheffer polynomials and integrable systems (Q2773104)

Operators of the form NEWLINE\[NEWLINE L=A(x,y)\partial_{xx}+2B(x,y)\partial_{xy}+C(x,y)\partial_{yy}+ D(x,y)\partial_{x}+E(x,y)\partial_{y}, NEWLINE\]NEWLINE where \(A(x,y),\dotsc,E(x,y)\) are polynomials in \(x\) and \(y\) with real coefficients are considered. Operator \(L\) is called admissible if for any positive integer \(n\) there exists \(n+1\) linearly independent polynomial eigenvalue solutions of degree \(n\geq 0\), \(LQ_n^{(i)}=\lambda_nQ_n^{(i)}\) and there are no polynomial solutions having degree less than \(n\) for the same value \(\lambda_n\). For an admissible operator \(L\) up to affine transformation there exist nine distinct types of \(L\) (see [\textit{G. K. Engelis}, Lato. Mat. Ezheg. 15, 169-202 (1974; Zbl 0302.33005) (Russian)]). In Theorem 2 for all nine types of classification schemes the algebraically independent operators \(I_1\) and \(I_2\) commuting with the operator \(L\) are given. In Theorem 3 (given without detailed proof) stated that existence of a nondegenerate orthogonality functional for an admissible operator \(L\) is equivalent to the existence of a second-order integral \(I\) commuting with \(L\): \([L,I]=0\).