ClearAll[f1, f2]
    f1 = Select[MemberQ[Tally /@ #, {_, 1}, All] &] @*Cases[{___,{(a_)..},___}] ;
    f2 = Pick[Cases[#, {___,{(a_)..},___}], MemberQ[#, {_, 1}, All]& /@ Map[Tally, #, {2}]] &;


**Examples:**

    lst1 = {{{4}}, {{7, 7}, {7, 7}}, {{5, 5}, {6, 6}}, {{2, 1}, {1, 1}}};
    f1 @ lst1
>    {{{4}}, {{2, 1}, {1, 1}}}

    f2 @ lst1
>    {{{4}}, {{2, 1}, {1, 1}}}


    lst2 = {{{3, 3}, {3, 2}}, {{1, 1, 1, 1}, {1, 2, 3, 0}, {1, 0, 0, 
         0}, {1, 0, 3, 2}}, {{1, 1, 1, 1, 1}, {1, 4, 3, 0, 4}, {1, 3, 3, 
         3, 0}, {1, 0, 3, 4, 4}, {1, 4, 0, 4, 4}}} ;
    f1 @ lst2 == f2 @lst2 ==lst2
>    True