Recently F. Pakovich and M. Muzychuk completely solved the vanishing problem for polynomial moments of the form \int_a^b P^k(x)Q(x) dP(x). This problem can be considered as an infinitesimal version of the Center-Focus problem for Abel differential equation, and the "moment centers" turn out to be pretty accurately described by certain composition relations between P and Q. For Laurent polynomials situation is more complicated. In a very recent work F. Pakovich has achieved a serious progress in understanding vanishing conditions for rational functions and, in particular, for Laurent polynomials. In particular, new relations with the Mathieu conjecture in representations of compact Lie groups have appeared, and (through the recent work of Wenhua Zhao) to certain questions closely related to the Jacobian conjecture.