Mathematica version 11.1.
The input
Solve[x (y + 1)^2 == 243 y, {x, y}, Integers]
yields the set of solutions
{{x -> 24, y -> 8}, {x -> 54, y -> 2}, {x -> 0, y -> 0}}
but there are also other solutions with negative x,y:
{x -> -486, y -> -2}, {x -> -108, y -> -4}, {x -> -30, y -> -10}
Notice that Element[-30, Integers] returns True.
