  We investigate degenerate cross-diffusion equations with a rank-deficient
diffusion matrix that are considered to model populations which move as to
avoid spatial crowding and have recently been found to arise in a mean-field
limit of interacting stochastic particle systems. To date, their analysis in
multiple space dimensions has been confined to the purely convective case with
equal mobility coefficients. In this article, we introduce a normal form for an
entropic class of such equations which reveals their structure of a symmetric
hyperbolic--parabolic system. Due to the state-dependence of the range and
kernel of the singular diffusive matrix, our way of rewriting the equations is
different from that classically used for symmetric second-order systems with a
nullspace invariance property. By means of this change of variables, we solve
the Cauchy problem for short times and positive initial data in
$H^s(\mathbb{T}^d)$ for $s>d/2+1$.
