  The L\'evy-Leblond equation with free potential admits a symmetry algebra
that is a $ \mathbb{Z}_2\times\mathbb{Z}_2 $-graded colour Lie superalgebra
(see arXiv:1609.08224). We extend this result in two directions by considering
a time-independent version of the L\'evy-Leblond equation. First, we construct
a $ \mathbb{Z}_2^3 $-graded colour Lie superalgebra containing operators that
leave the eigenspaces invariant and demonstrate the utility of this algebra in
constructing general solutions for the free equation. Second, we find that the
ladder operators for the harmonic oscillator generate a $
\mathbb{Z}_2\times\mathbb{Z}_2 $-graded colour Lie superalgebra and we use the
operators from this algebra to compute the spectrum. These results illustrate
two points: the L\'evy-Leblond equation admits colour Lie superalgebras with
gradings higher than $ \mathbb{Z}_2\times\mathbb{Z}_2 $ and colour Lie
superalgebras appear for potentials besides the free potential.
