In this paper we perform a multiple scales analysis on the coupled KdV systems associated with some matrix Schr\"odinger operators. We derive the corresponding matrix NLS equations, together with their zero-curvature respresentations. One particular class of these constitutes a new, integrable NLS system.

We then consider the stationary flows of these equations and present a new integrable generalisation of the Garnier system, having a $2N \times 2N$ Lax matrix, from which we calculate the constants of motion.