  The space of solutions of the exact renormalization group fixed point
equations of the two-dimensional $RP^{N-1}$ model, which we recently obtained
within the scale invariant scattering framework, is explored for continuous
values of $N\geq 0$. Quasi-long-range order occurs only for $N=2$, and allows
for several lines of fixed points meeting at the BKT transition point. A rich
pattern of fixed points is present below $N^*=2.24421..$, while only zero
temperature criticality in the $O(N(N+1)/2-1)$ universality class can occur
above this value. The interpretation of an extra solution at $N=3$ requires the
identitication of a path to criticality specific to this value of $N$.
