  The linear $\sigma$-model with a chemical potential for hypercharge is a toy
model for the description of the dynamics of the kaon condensate in high
density QCD. We analyze the dynamics of the gauged version of this model. It is
shown that spontaneous breakdown of $SU(2) \times U(1)_{Y}$ symmetry, caused by
the chemical potential, is always accompanied by spontaneous breakdown of both
rotational symmetry and electromagnetic $U(1)_{em}$. The spectrum of
excitations in this model is rich and, because of rotational symmetry
breakdown, anisotropic. It is shown that there exist excitation branches that
behave as phonon like quasiparticles for small momenta and as roton like ones
for large momenta. This suggests that this model can be relevant for
anisotropic superfluid systems.
