  We study the zero energy modes that arise in an unusual vortex configuration
involving both the kinetic energy and an appropriate mass term in a model which
exhibits birefringent Dirac fermions as its low energy excitations. We find the
surprising feature that the ratio of the length scales associated with states
centered on vortex and anti-vortex topological defects can be arbitrarily
varied but that fractionalization of quantum numbers such as charge is
unaffected. We discuss this situation from a symmetry point of view and present
numerical results for a specific lattice model realization of this scenario.
