  We study the $O(N)$ model in dimension three (3$d$) at large and infinite $N$
and show that the line of fixed points found at $N=\infty$ --the
Bardeen-Moshe-Bander (BMB) line-- has an intriguing origin at finite $N$. The
large $N$ limit that allows us to find the BMB line must be taken on particular
trajectories in the $(d,N)$-plane: $d=3-\alpha/N$ and not at fixed dimension
$d=3$. Our study also reveals that the known BMB line is only half of the true
line of fixed points, the second half being made of singular fixed points. The
potentials of these singular fixed points show a cusp for a finite value of the
field and their finite $N$ counterparts a boundary layer.
