  We study the structure of vortex solutions in a Ginzburg-Landau system for
two complex valued order parameters. We consider the Dirichlet problem in the
disk in R^2 with symmetric, degree-one boundary condition, as well as the
associated degree-one entire solutions in all of R^2. Each problem has
degree-one equivariant solutions with radially symmetric profile vanishing at
the origin, of the same form as the unique (complex scalar) Ginzburg-Landau
minimizer. We find that there is a range of parameters for which these
equivariant solutions are the unique locally energy minimizing solutions for
the coupled system. Surprisingly, there is also a parameter regime in which the
equivariant solutions are unstable, and minimizers must vanish separately in
each component of the order parameter.
