  Conformal Galilei Algebras labeled by $d,\ell$ (where $d$ is the number of
space dimensions and $\ell$ denotes a spin-${\ell}$ representation w.r.t. the
$\mathfrak{sl}(2)$ subalgebra) admit two types of central extensions, the
ordinary one (for any $d$ and half-integer $\ell$) and the exotic central
extension which only exists for $d=2$ and ${\ell}\in\mathbb{N}$.
  For both types of central extensions invariant second-order PDEs with
continuous spectrum were constructed in [1]. It was later proved in [2] that
the ordinary central extensions also lead to oscillator-like PDEs with discrete
spectrum.
  We close in this paper the existing gap, constructing \textcolor{black}{a new
class of second-order invariant PDEs for the exotic centrally extended CGAs;
they admit a discrete and bounded spectrum when applied to a lowest weight
representation. These PDEs are markedly different with respect to their
ordinary counterparts. The ${\ell}=1$ case (which is the prototype of this
class of extensions, just like the $\ell=\frac{1}{2}$ Schr\"odinger algebra is
the prototype of the ordinary centrally extended CGAs) is analyzed in detail.
