  We analytically calculate the ground state pairing symmetry and excitation
spectra of two holes doped into the half-filled t-t'-t''-J_z model in the
strong-coupling limit (J_z >> |t|, |t'|, |t''|). For the t'-t''-J_z model,
there are regions of d-wave, s-wave, and p-wave symmetry. We find that the
t-J_z model maps in lowest order onto the t'-t''-J_z model on the boundary
between d and p symmetry, with a flat lower branch of the pair excitation
spectrum. In higher order d-wave symmetry is selected; however, we predict that
the addition of the appropriate t' and/or t'' should drive the hole-pair
symmetry to p-wave. We perturbatively construct an extended quasi-pair for the
t-J_z model. We compare with analytic calculations for a 2x2 plaquette and
numerical work, and discuss implications for the experimentally relevant
parameter regime.
