Dear Mr. Lichtblau,

Thank you for your response.

I am also grateful for the reference to the matix pencil, may be it will be helpful for solution of the problem.

In the example I uploaded [H,TotSz]=0, on the other hand as I presented in the end of the file, one of the eigenvactors of H is e.g.

{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.0105979, 0, 0.0102231, -0.706877, 0, 0, 0, 0, -0.020821, 0.706877, 0, 0, 0}

while the action of TotSz on the same column yields

{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -0.0105979, 0, -0.0102231, 1.41375, 0, 0, 0, 0, 0.020821, -1.41375, 0, 0, 0}

As you can see elements of the function are not uniformally multiplied by the same number (i.e. by the eigenvalue of the TotSz corresponding to this "eigen"-function).

Take for example the 15-th element which is multiplied by -1, and 18-th element which is multiplied by -2.