  An action with $n$ parameters, which generalizes the $O(N) - R P^{N-1}$
-model, is considered in one dimension for general $N$. We use asymptotic
expansion techniques to determine where the model becomes critical and show
that for the actions considered there exists a family of hypersurfaces whose
asymptotic behaviour determines a one-parameter family of new universality
classes. They interpolate between the $O(N)$-vector-model-class and the $R
P^{N-1}$-model-class. Furthermore continuum limits are discussed, including the
exceptional case $N=2$.
