This is mainly a reference request. Let us define a parameterized expression on a finite alphabet $\Sigma$ as follows:
$$e,e':= w\mid w^i \mid e\cdot e'$$
Where $w\in\Sigma^+$ is a word, and $i$ is an abstract parameter, i.e. an integer variable.
A parameterized equation is an equation $e=e'$, where $e,e'$ are as above, and every integer variable is used at most once overall.
We want an algorithm outputting the set of solutions as a boolean combination of equations on the integer parameters.
Example: $a^i b (ab)^j=(ab)^k$.
Here we expect the algorithm to answer: "$i=1$ and $k=j+1$"
This seems to be done in more general cases, e.g. by  which deals with word variables, as most papers I can find when searching for "word equations". But I expect that this simpler case is a lot easier to handle. Unfortunately it is hard to find references, probably because I don't know the correct keywords to search for this. I think I can solve the problem by hand but would like to know relevant references on the subject.
Question 1: Do you know of a source describing a simple algorithm for this problem ?
Question 2: What happens if we allow disjunctions of words under iterations, i.e equations like $(ab+ba+bab)^i=b^j(ab)^k$ ? Is there in the literature an algorithm describing the set of solutions for $i,j,k$ ?
 Wojciech Plandowski. 2006. An efficient algorithm for solving word equations. In Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing (STOC '06). Association for Computing Machinery, New York, NY, USA, 467–476. DOI:https://doi.org/10.1145/1132516.1132584