On commuting matrix differential operators (Q5954311)

For a given differential expression \(L\), consider a problem of finding a differential expression \(P\) satisfying \([P,L]=0\) other than one such that both \(P\) and \(L\) are in \(\mathbb{C}[A]\) for some differential expression \(A.\) This problem has drawn much attention, as well as its relationship with completely integrable systems of partial differential equations. Here, the author treats the differential expression NEWLINE\[NEWLINE L=Q_0\frac{d^n}{dx^n}+\cdots + Q_n, NEWLINE\]NEWLINE where \(Q_j\), \(j=0,1,\dots, n\), are \(m\times m\)-matrices, \(mn>1\), with rational or simply periodic entries. Under certain conditions, he proves the following: if, for all \(z\in\mathbb{C},\) all solutions to \(Ly=zy\) are meromorphic, then there exists a differential expression \(P\) satisfying \([P,L]=0.\) NEWLINENEWLINENEWLINEThe results for \(L=J d/dx+Q,\) where \(J\) and \(Q\) are \(2\times 2\)-matrices, are applied to the AKNS hierarchy, and all rational and simply periodic algebro-geometric AKNS potentials are charactereized.