You can ask for integer solutions in Mathematica with

> Solve[x/(y + z) + y/(z + x) + z/(x + y) == 4, {x, y, z}, Integers]

but it gives a very long output in terms of Root[]s and conditional expressions.

I initially tried that on WolframAlpha but it doesn't understand Mathematica syntax. But one can enter it on https://lab.wolframcloud.com after creating an account.

Thanks. At least I could check the solution there and it is correct.

I got "Standard computation time exceeded...".