  We study pattern formation associated with the polarization degree of freedom
of the electric field amplitude in a mean field model describing a nonlinear
Kerr medium close to a two-photon resonance, placed inside a ring cavity with
flat mirrors and driven by a coherent $\hat x$-polarized plane-wave field. In
the self-focusing case, for negative detunings the pattern arises naturally
from a codimension two bifurcation. For a critical value of the field intensity
there are two wave numbers that become unstable simultaneously, corresponding
to two Turing-like instabilities. Considered alone, one of the instabilities
would originate a linearly polarized hexagonal pattern whereas the other
instability is of pure vectorial origin and would give rise to an elliptically
polarized stripe pattern. We show that the competition between the two
wavenumbers can originate different structures, being the detuning a natural
selection parameter.
