  We study deformations of the harmonic oscillator algebra known as polynomial
Heisenberg algebras (PHAs), and establish a connection between them and
extended affine Weyl groups of type $A^{(1)}_m$, where $m$ is the degree of the
PHA. To establish this connection, we employ supersymmetric quantum mechanics
to first connect a polynomial Heisenberg algebra to symmetric systems of
differential equations. This connection has been previously used to relate
quantum systems to non-linear differential equations; most notably, the fourth
and fifth Painlev\'e equations. Once this is done, we use previous studies on
the B\"acklund transformations of Painlev\'e equations and generalizations of
their symmetric forms characterized by extended affine Weyl groups. This work
contributes to better understand quantum systems and the algebraic structures
characterizing them.
