The following problem on Diophantine equation is still solved or not I don't know. However, I found few solutions by trail and error method.

Problem: $X^2 - X = Y^5 - Y$ has integer solutions or not?

Solution: As part of my regular involvement on this problem, I have found the following set of integer solutions by trail and error method or inspection method.

i.e., $(0, -1), (1, -1), (0, 0) =$ trivial solution, $(1, 0),$ $(0, 1)$ and $(1, 1)$.

If there exits still more solutions apart from above, HOW TO FIND SUCH SOLUTIONS? Importantly, is there any method to find solutions instead of trail or inspection method?

with regards, mahoob