  We use scale invariant scattering theory to obtain the exact equations
determining the renormalization group fixed points of the two-dimensional
$CP^{N-1}$ model, for $N$ real. Also due to special degeneracies at $N=2$ and
3, the space of solutions for $N\geq 2$ reduces to that of the $O(N^2-1)$
model, and accounts for a zero temperature critical point. For $N<2$ the space
of solutions becomes larger than that of the $O(N^2-1)$ model, with the
appearance of new branches of fixed points relevant for criticality in gases of
intersecting loops.
