This paper is about tracking multiple targets with the so-called Symmetric Measurement Equation (SME) filter. The SME filter uses symmetric functions, e.g., symmetric polynomials, in order to remove the data association uncertainty from the measurement equation. By this means, the data association problem is converted to a nonlinear state estimation problem. In this work, an efficient optimal Gaussian filter based on analytic moment calculation for discrete-time multi-dimensional polynomial systems corrupted with Gaussian noise is derived, and then applied to the polynomial system resulting from the SME filter. The performance of the new method is compared to an UKF implementation by means of typical multiple target tracking scenarios.