On mep-relations in the wreath product of groups

Let a, b be two long cycles in an alternating group An, satisfying relations a=[a,kb] and b=[b,ka]. We show that every pair of elements of the form x=(X,a), y=(Y,b), where the sum of coefficients of X and Y is equal zero, satisfies relations x=[x,ly], y=[y,lx] in the wreath product ′(Sn≀Zm) for m coprime with n and for an l divisible by k. We show also that for n=5,7,13 and for m coprime with n, ′(Sn≀Zm) is generated by such pairs.