  The dynamical symmetries of $1+1$-dimensional Matrix Partial Differential
Equations with a Calogero potential (with/without the presence of an extra
oscillatorial De Alfaro-Fubini-Furlan, DFF, damping term) are investigated. The
first-order invariant differential operators induce several invariant algebras
and superalgebras. Besides the $sl(2)\oplus u(1)$ invariance of the Calogero
Conformal Mechanics, an $osp(2|2)$ invariant superalgebra, realized by
first-order and second-order differential operators, is obtained. The invariant
algebras with an infinite tower of generators are given by the universal
enveloping algebra of the deformed Heisenberg algebra, which is shown to be
equivalent to a deformed version of the Schr\"odinger algebra. This vector
space also gives rise to a higher spin (gravity) superalgebra. We furthermore
prove that the pure and DFF Matrix Calogero PDEs possess isomorphic dynamical
symmetries, being related by a similarity transformation and a redefinition of
the time variable.
