  We analytically calculate the ground state pairing symmetry and excitation
spectra of two holes doped into the half-filled t-t'-t"-Jz model in the
strong-coupling limit (Jz>>|t|,|t'|,|t"|). In leading order, this reduces to
the t'-t"-Jz model, where there are regions of d-wave, s-wave, and (degenerate)
p-wave symmetry. We find that the t-Jz model maps in lowest order onto the
t'-t"-Jz model on the boundary between d and p symmetry, with a flat lower band
in the pair excitation spectrum. In higher order, d-wave symmetry is selected
from the lower pair band. However, we observe that the addition of the
appropriate t'<0 and/or t''>0, the signs of t' and t" found in the hole-doped
cuprates, could drive the hole-pair symmetry to p-wave, implying the
possibility of competition between p-wave and d-wave pair ground states. (An
added t'>0 and/or t" <0 generally tend to promote d-wave symmetry.) We
perturbatively construct an extended quasi-pair for the t-Jz model. In leading
order, there are contributions from sites at a distance of sqrt{2} lattice
spacings apart; however, contributions from sites 2 lattice spacings apart,
also of the same order, vanish identically. Finally, we compare our approach
with analytic calculations for a 2x2 plaquette and with existing numerical
work, and discuss possible relevance to the physical parameter regime.
