  A type of electron pairing model with spin-orbit interactions or Zeeman
coupling is solved exactly in the framework of Richardson ansatz. Based on the
exact solutions for the case with spin-orbit interactions, it is shown
rigorously that the pairing order parameter has always the $p+ip$-wave symmetry
regardless of the strength of pairing interaction, justifying an important
conclusion obtained by the mean field theory. Analytical results are derived
for two simple systems with respectively spin-orbit interactions and Zeeman
coupling.
