On the intrinsic structure of the solution set to the Yang–Baxter-like matrix equation

In this paper we analyze isolated and connected points of the solution set to the Yang–Baxter-like matrix equation A X A = X A X . In particular, if A is a regular matrix, we prove that the trivial solution X = 0 is always isolated in the solution set, while the trivial solution X = A is isolated under some natural conditions, and an example shows that these conditions cannot be omitted. Conversely, we demonstrate that the two trivial solutions can be path-connected in the solution set when A is singular. Furhter, we prove that every nontrivial non-commuting solution is always contained in some path-connected subset of the solution set, regardless of whether A is regular or singular. Additionally, we develop new methods for obtaining infinitely many new nontrivial non-commuting solutions (for both regular and singular A ). Explicit examples are provided after almost every theoretical result.