  This paper investigates an inhomogeneous non-zero-sum linear-quadratic (LQ,
for short) differential game problem whose state process and cost functional
are regulated by a Markov chain. Under the $L^2$ stabilizability framework, we
first provide a sufficient condition to ensure the $L^2$-integrability of the
state process and study a class of linear backward stochastic differential
equation (BSDE, for short) in infinite horizon. Then, we seriously discuss the
LQ problem and show that the closed-loop optimal control is characterized by
the solutions to coupled algebra Riccati equations (CAREs, for short) with some
stabilizing conditions and a linear BSDE. Based on those results, we further
analyze the non-zero-sum stochastic differential game problem and give the
closed-loop Nash equilibrium through the solution to a system of two
cross-coupled CAREs and two cross-coupled BSDEs. Finally, some related
numerical
