  The $^3P_2$-$^3F_2$ pairing model is generally considered to provide an
adequate description of the superfluid states of neutron matter at densities
some 2-3 times that of saturated symmetrical nuclear matter. The problem of
solving the system of BCS gap equations expressing the $^3P_2$-$^3F_2$ model is
attacked with the aid of the separation approach. This method, developed
originally for quantitative study of S-wave pairing in the presence of strong
short-range repulsions, serves effectively to reduce the coupled, singular,
nonlinear BCS integral equations to a set of coupled algebraic equations. For
the first time, sufficient precision becomes accessible to resolve small energy
splittings between the different pairing states. Adopting a perturbative
strategy, we are able to identify and characterize the full repertoire of real
solutions of the $^3P_2$-$^3F_2$ pairing model, in the limiting regime of small
tensor-coupling strength. The P-F channel coupling is seen to lift the striking
parametric degeneracies revealed by a earlier separation treatment of the pure,
uncoupled $^3P_2$ pairing problem. Remarkably, incisive and robust results are
obtained solely on the basis of analytic arguments. Unlike the traditional
Ginzburg-Landau approach, the analysis is not restricted to the immediate
vicinity of the critical temperature, but is equally reliable at zero
temperature. Interesting connections and contrasts are drawn between triplet
pairing in dense neutron matter and triplet pairing in liquid $^3$He.
