  We consider two-dimensional non-centrosymmetric superconductors, where the
order parameter is a mixture of s-wave and p-wave parts, in the presence of an
externally induced Zeeman splitting. We derive the conditions under which the
system is in a non-Abelian phase. By considering the non-degenerate zero-energy
Majorana solutions of the Bogoliubov-de Gennes (BdG) equations for a vortex and
by constructing a topological invariant, we show that the condition for the
non-Abelian phase to exist is completely independent of the triplet pairing
amplitude. The existence condition for the non-Abelian phase derived from the
real space solutions of the BdG equations involves the Pfaffian of the BdG
Hamiltonian at k = 0, which is completely insensitive to the magnitude of the
p-wave component of the order parameter. We arrive at the same conclusion by
using the appropriate topological invariant for this case. This is in striking
contrast to the analogous condition for the time-reversal invariant topological
phases, in which the amplitude of the p-wave component must be larger than the
amplitude of the s-wave piece of the order parameter. As a by-product, we
establish the intrinsic connection between the Pfaffian of the BdG Hamiltonian
at k = 0 (which arises at the BdG approach) and the relevant Z topological
invariant.
