Solve an equaltion similar to Sylvester equation

Question

Suppose $A,B\in C^{N\times N}$ with their eigenvalues $\lambda_1,...,\lambda_N$ and $\mu_1,...,\mu_N$ respectively.

Please give the sufficient and necessary conditions of that: there exists non-zero $X$ for this equation:

$A^2X+XB^2-2AXB=O$

Answer 1

Hint: rewrite the equation as $$ A(\underbrace{AX-XB}_{Y})-(\underbrace{AX-XB}_{Y})B=0. $$ Can it happen that it has only the trivial solution $Y=0$?