  We give an algebraic criterion for the existence of projectively
Hermitian-Yang-Mills metrics on a holomorphic vector bundle $E$ over some
complete non-compact K\"ahler manifolds $(X,\omega)$, where $X$ is the
complement of a divisor in a compact K\"ahler manifold and we impose some
conditions on the cohomology class and the asymptotic behaviour of the K\"ahler
form $\omega$. We introduce the notion of stability with respect to a pair of
$(1,1)$-classes which generalizes the standard slope stability. We prove that
this new stability condition is both sufficient and necessary for the existence
of projectively Hermitian-Yang-Mills metrics in our setting.
