Pushnitskis $mu$-invariant and Schrodinger operators with embedded eigenvalues

In this note, under a certain assumption on an affine space of operators, which admit embedded eigenvalues, it is shown that the singular part of the spectral shift function of any pair of operators from this space is an integer-valued function. The proof uses a natural decomposition of Pushnitskis $mu$-invariant into absolutely continuous and singular parts. As a corollary, the Birman-Krein formula follows.