$a^m + b^n = c^{m+n}$ general solution over integers

Question

It is a follow-up to this question which asked if any solution to

$$a^m + b^n = c^{m+n}$$

exists over the integers.

The post contains some of the elaborations on this achieved by me and others.

I am looking for a general solution.

Since there can be many families of solutions, I'm looking for some sort of general description of those families.
Also you're welcome to provide new families of solutions, as finding all of them is indeed a difficult task.

Answer 1

COMMENT.- I feel it is to much asked. Anyway it cannot be rationally parameterized with polynomials like as for Pythagorean triples. In fact if $$a=f(x,y), b=g(x,y), c=h(x,y)$$ with $$f^m(x,y)+g^n(x,y)=h^{m+n}(x,y)$$ where $f,g,h\in\mathbb Z[x,y]$, then fixing, for example $y=3$, or any other integer, we get an equality of univariate polynomials $$p_1^m(x)+p_2^n(x)=p_3^{m+n}(x)$$ valid for more values of $x$ than the higher degree of $p_1,p_2,p_3$ from which we can deny the aforementioned possibility.