The author discusses an isospectral problem, using an extension of the theorem of Boiti-Pempinelli-Tu concerning the gradient of conserved densities of integrable systems. Let (1) be an isospectral problem where is an matrix depending on the spectral parameter and the potential . A matrix with is constructed. Then set , where , and the gradient of H with the transpose of U. It is well known that for (2) where for arbitrary constants
The author proves: Theorem 1. Let be given as above. Then one has (3) (, where is the gradient of defined by [see D. H. Sattinger, Stud. Appl. Math. 72, 65-86 (1985; Zbl 0584.58022)];
Theorem 2. Let the conserved density h be given by . Then one has also the equation (3).
For the isospectral problem, the author obtains:
Theorem 3. The gradient of eigenvalues of the isospectral problem (1) satisfies the stationary zero-curvature equation .