same parity.

\[remark2\] One can consider the interpolation problem (\[2n-interpolation\]) with $O(z^{M})$, where $M>2n$ or $M<2n$, instead of $O(z^{2n})$. Then we accordingly get overdetermined and underdetermined moment systems of the type (\[SRS\]): $$\label{SRS_M} \sum_{k=1}^n \mu_k \lambda_k^m=s_m,\qquad
m=\overline{0,M-1}.$$ In some cases the process of solving the consistent systems (\[SRS\_M\]) with $M\ne 2n$ can be reduced to the one of the standard systems (\[SRS\]) (with $M=2n$). It can be done by eliminating the superfluous equations or adding the missed ones (regarding this see also [@Lyubich §5]). But in the present paper we do not consider the case $M\ne 2n$ for the following reasons.

The overdetermined systems (\[SRS\_M\]) belong to the non-regular problems of the form (\[SRS\]), where $2n$ is exchanged by $M$ and $\mu_{k}=0$ for $k{\geqslant}n+1$. To apply the Prony-Sylvester method or some other analytical approaches in the standard subsystem of the system (\[SRS\_M\]), one needs a preliminary analysis of its consistency. As far as we know, there exist no reasonably general methods for this purpose. Moreover, it can be seen from the corresponding overdetermined interpolation problem of the form (\[2n-interpolation\]), where $O(z^{2n})$ is exchanged by $O(z^{M})$, that its consistency is quite rigidly connected with individual properties of the functions $f$ and 