  The separation method developed earlier by us [Nucl. Phys. {\bf A598} 390
(1996)] to calculate and analyze solutions of the BCS gap equation for
$^1$S$_0$ pairing is extended and applied to $^3$P$_2$--$^3$F$_2$ pairing in
pure neutron matter. The pairing matrix elements are written as a separable
part plus a remainder that vanishes when either momentum variable is on the
Fermi surface. This decomposition effects a separation of the problem of
determining the dependence of the gap components in a spin-angle representation
on the magnitude of the momentum (described by a set of functions independent
of magnetic quantum number) from the problem of determining the dependence of
the gap on angle or magnetic projection. The former problem is solved through a
set of nonsingular, quasilinear integral equations, providing inputs for
solution of the latter problem through a coupled system of algebraic equations
for a set of numerical coefficients. An incisive criterion is given for finding
the upper critical density for closure of the triplet gap. The separation
method and its development for triplet pairing exploit the existence of a small
parameter, given by a gap-amplitude measure divided by the Fermi energy. The
revised BCS equations admit analysis revealing universal properties of the full
set of solutions for $^3$P$_2$ pairing in the absence of tensor coupling,
referring especially to the energy degeneracy and energetic order of these
solutions. The angle-average approximation introduced by Baldo et al. is
illuminated in terms of the separation-transformed BCS problem and the small
parameter expansion...
