Main Question or Discussion Point
Need to find all the posible solutions (a,b,c) for 1/a + 1/b + 1/c = 1, a,b,c integers.
That is true only if a, b, c are restricted to the positive integers. The OP just said integers.You can start by assuming, for the time being, that [itex]a\le b\le c[/itex] -- you can rearrange the terms later if you need to.
If a = 1 then the sum is too large