LetKbe a field with charK≠ 3 and it two positive integers such that 1 ≤i<t/2,t≠ 3i. The classification problem for maximal Cohen-Macaulay modules overK[[X,Y]]/(Xt+Y3) is complicated ift≥ 6, because there exist parameter families of non-isomorphic maximal Cohen-Macaulay modules [Sc], or [GK], [Yo, Ch.9] and [DG]). Here we describe parameter families of such modulesN, such that N/YN is a direct sum of copies ofK[[X]]/(Xi)K[[X]]/(Xt-i).