  For the case of the single-O($N$)-vector linear sigma models the critical
behaviour following from any $A_k$ singularity in the action is worked out in
the double scaling limit $N \rightarrow \infty$, $f_r \rightarrow f_r^c$, $2
\leq r \leq k$. After an exact elimination of Gaussian degrees of freedom, the
critical objects such as coupling constants, indices and susceptibility matrix
are derived for all $A_k$ and spacetime dimensions $0 \leq D < 4$. There appear
exceptional spacetime dimensions where the degree $k$ of the singularity $A_k$
is more strongly constrained than by the renormalizability requirement.
