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The first-order differential L\'evy-Leblond equations (LLE's) are thenon-relativistic analogs of the Dirac equation, being square roots of($1+d$)-dimensional Schr\"odinger or heat equations. Just like the Diracequation, the LLE's possess a natural supersymmetry. In previous works it wasshown that non supersymmetric PDE's (notably, the Schr\"odinger equations forfree particles or in the presence of a harmonic potential), admit a natural${\mathbb Z}_2$-graded Lie symmetry. In this paper we show that, for a certainclass of supersymmetric PDE's, a natural ${\mathbb Z}_2\times{\mathbbZ}_2$-graded Lie symmetry appears. In particular, we exhaustively investigatethe symmetries of the $(1+1)$-dimensional L\'evy-Leblond Equations, both in thefree case and for the harmonic potential. In the free case a ${\mathbbZ}_2\times{\mathbb Z}_2$-graded Lie superalgebra, realized by first andsecond-order differential symmetry operators, is found. In the presence of anon-vanishing quadratic potential, the Schr\"odinger invariance is maintained,while the ${\mathbb Z}_2$- and ${\mathbb Z}_2\times{\mathbb Z}_2$- gradedextensions are no longer allowed. The construction of the ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie symmetry of the ($1+2$)-dimensional free heat LLEintroduces a new feature, explaining the existence of first-order differentialsymmetry operators not entering the super Schr\"odinger algebra.

${\mathbb Z}_2\times {\mathbb Z}_2$-graded Lie Symmetries of the L\'evy-Leblond Equations