In this paper, we examine in detail the principal branches of solutions that
arise in vector discrete models with nonlinear inter-component coupling and
four wave mixing. The relevant four branches of solutions consist of two single
mode branches (transverse electric and transverse magnetic) and two mixed mode
branches, involving both components (linearly polarized and elliptically
polarized). These solutions are obtained explicitly and their stability is
analyzed completely in the anti-continuum limit (where the nodes of the lattice
are uncoupled), illustrating the supercritical pitchfork nature of the
bifurcations that give rise to the latter two, respectively, from the former
two. Then the branches are continued for finite coupling constructing a full
two-parameter numerical bifurcation diagram of their existence. Relevant
stability ranges and instability regimes are highlighted and, whenever
unstable, the solutions are dynamically evolved through direct computations to
monitor the development of the corresponding instabilities. Direct connections
to the earlier experimental work of Meier et al. [Phys. Rev. Lett. {\bf 91},
143907 (2003)] that motivated the present work are given.Comment: 13 pages, 10 figure
