  We introduce and investigate the class of central potentials
$$V_{\text{CIC}}(g^{2},\mu^{2},\ell,R;r)=-\frac{g^{2}}{R^{2}}
(\frac{r}{R})^{4\ell} {[ 1+(\frac{1}{2\ell+1})
(\frac{r}{R})^{2\ell+1}]^{2}-1+\mu^{2}}^{-2}$$, which possess, in the context
of nonrelativistic quantum mechanics, a number of $\ell$-wave bound states
given by the ($\ell$-independent !) formula $$N_{\ell}^{\text{(CIC)}}(g^{2},\mu
^{2}) ={{\frac{1}{\pi}\sqrt{g^{2}+\mu^{2}-1} (\sqrt{\mu^{2}-1})^{-1}
\arctan(\sqrt{\mu^{2}-1})}}$$. Here $g$ and $\mu$ are two arbitrary real
parameters, $\ell$ is the angular momentum quantum number, and the double
braces denote of course the integer part. An extension of this class features
potentials that possess the same number of $\ell$-wave bound states and behave
as $(a/r)^{2}$ both at the origin ($r\to 0^{+}$) and at infinity ($r\to
\infty$), where $a$ is an additional free parameter.
