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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
True, I missed that. There is one family of solutions which includes negative numbers.That is true only if a, b, c are restricted to the positive integers. The OP just said integers.
If two are negative you have 1 - 1/a - 1/b < 1. If three are negative you have -1/a - 1/b - 1/c < 0.Do you think there are more?