  In this article, a system of Yang-Baxter-type matrix equations is studied,
$XAX=BXB$, $XBX=AXA$, which "generalizes" the matrix Yang-Baxter equation and
exhibits a broken symmetry. We investigate the solutions of this system from
various geometric and topological points of view. We analyze the existence of
doubly stochastic solutions and intertwining solutions to the system and
describe the conditions for their existence. Furthermore, we characterize the
case when $A$ and $B$ are idempotent orthogonal complements. i.e., $A^2 =A,
B^2= B, AB = BA =0$. We also completely characterize the set of solutions for
$n=2$ using commutative algebraic techniques.
