  In this paper, we establish some necessary and sufficient conditions for the
existence of solutions to the system of operator equations $ BXA=B=AXB $ in the
setting of bounded linear operators on a Hilbert space, where the unknown
operator $X$ is called the inverse of $A$ along $B$. After that, under some
mild conditions we prove that an operator $X$ is a solution of $ BXA=B=AXB $ if
and only if $B \stackrel{*}{ \leq} AXA$, where the $*$-order $C\stackrel{*}{
\leq} D$ means $CC^*=DC^*, C^*C=C^*D$. Moreover we present the general solution
of the equation above. Finally, we present some characterizations of $C
\stackrel{*}{ \leq} D$ via other operator equations.
