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We construct, for any given ${\ell}=\frac{1}{2}+{\mathbb{N}}_0$, thesecond-order, linear PDEs which are invariant under the centrally extendedConformal Galilei Algebra. \par At the given ${\ell}$, two invariant equationsin one time and ${\ell}+\frac{1}{2}$ space coordinates are obtained. The firstequation possesses a continuum spectrum and generalizes the free Schr\"odingerequation (recovered for ${\ell}=\frac{1}{2}$) in $1+1$ dimension. The secondequation (the "$\ell$-oscillator") possesses a discrete, positive spectrum. Itgeneralizes the $1+1$-dimensional harmonic oscillator (recovered for$\ell=\frac{1}{2}$). The spectrum of the ${\ell}$-oscillator, derived from aspecific $osp(1|2\ell+1)$ h.w.r., is explicitly presented.\par The two sets ofinvariant PDEs are determined by imposing (representation-dependent) {\iton-shell invariant conditions} both for {\it degree} $1$ operators (those withcontinuum spectrum) and for {\it degree } $0$ operators (those with discretespectrum).\par The on-shell condition is better understood by enlarging theConformal Galilei Algebras with the addition of certain second-orderdifferential operators. Two compatible structures (the algebra/superalgebraduality) are defined for the enlarged set of operators.

${\ell}$-oscillators from second-order invariant PDEs of the centrally extended Conformal Galilei Algebras