lst = {{{4}}, {{7, 7}, {7, 7}}, {{5, 5}, {6, 6}}, {{2, 1}, {1, 1}}};

    Pick[#, MemberQ[#, {_, 1}, 2] & /@ Map[Tally, #, {2}]] & @ lst
>    {{{4}}, {{2, 1}, {1, 1}}}