  We study the the Dirichlet problem for the cross-diffusion system \[
\partial_tu_i=\operatorname{div}\left(a_iu_i\nabla
(u_1+u_2)\right)+f_i(u_1,u_2),\quad i=1,2,\quad a_i=const>0, \] in the cylinder
$Q=\Omega\times (0,T]$. The functions $f_i$ are assumed to satisfy the
conditions $f_1(0,r)=0$, $f_2(s,0)=0$, $f_1(0,r)$, $f_2(s,0)$ are locally
Lipschitz-continuous. It is proved that for suitable initial data $u_0$, $v_0$
the system admits segregated solutions $(u_1,u_2)$ such that $u_i\in
L^{\infty}(Q)$, $u_1+u_2\in C^{0}(\overline{Q})$, $u_1+u_2>0$ and $u_1\cdot
u_2=0$ everywhere in $Q$. We show that the segregated solution is not unique
and derive the equation of motion of the surface $\Gamma$ which separates the
parts of $Q$ where $u_1>0$, or $u_2>0$. The equation of motion of $\Gamma$ is a
modification of the Darcy law in filtration theory. Results of numerical
simulation are presented.
