  We introduce and prove a one-parameter refinement of the Razumov-Stroganov
correspondence. This is achieved for fully-packed loop configurations (FPL) on
domains which generalize the square domain, and which are endowed with the
gyration operation. We consider one given side of the domain, and FPLs such
that the only straight-line tile on this side is black. We show that the
enumeration vector associated to such FPLs, weighted according to the position
of the straight line and refined according to the link pattern for the black
boundary points, is the ground state of the scattering matrix, an integrable
one-parameter deformation of the O(1) Dense Loop Model Hamiltonian. We show how
the original Razumov-Stroganov correspondence, and a conjecture formulated by
Di Francesco in 2004, follow from our results.
