  We investigate the general class of second-order PDEs, invariant under the
$d=1$ $\ell=\frac{1}{2}+{\mathbb N}_0$ centrally extended Conformal Galilei
Algebras, pointing out that they are deformations of decoupled systems. For
$\ell=\frac{3}{2}$ the unique deformation parameter $\gamma$ belongs to the
fundamental domain $\gamma\in ]0,+\infty[$. We show that, for any $\gamma\neq
0$, invariant PDEs with discrete spectrum (either bounded or unbounded) induce
cryptohermitian operators possessing the same spectrum as two decoupled
oscillators, provided that their frequencies are in the special ratio
$r=\frac{\omega_2}{\omega_1}=\pm\frac{1}{3},\pm 3$ (the negative energy
solutions correspond to a special case of Pais-Uhlenbeck oscillator), where
$\omega_1,\omega_2$ are two different parameters of the invariant PDEs. We also
consider the $\gamma=0$ decoupled system for any value $r$ of the ratio. It
possesses enhanced symmetry at the critical values $r=\pm \frac{1}{3}, \pm
1,\pm 3$. Two inequivalent $12$-generator symmetry algebras are found at $r
=\pm\frac{1}{3},\pm 3$ and $r=\pm 1$, respectively. The $\ell=\frac{3}{2}$
Conformal Galilei Algebra is not a subalgebra of the decoupled symmetry
algebra. Its $\gamma\rightarrow 0$ contraction corresponds to a $8$-generator
subalgebra of the decoupled $r=\pm\frac{1}{3},\pm 3$ symmetry algebra. The
features of the $\ell\geq \frac{5}{2}$ invariant PDEs are briefly discussed.
