MR2100341 Carballo, Juan M. ; Fernández C., David J. ; Negro,
Javier ; Nieto, Luis M. Polynomial Heisenberg algebras. J. Phys. A
37 (2004), no. 43, 10349--10362.



In textbooks on quantum mechanics, one of the most popular
algebraic interpretations of the equidistance of the spectrum of
the harmonic oscillator is often presented as a consequence of the
factorization of its Hamiltonian $H = p^2+q^2$ into a product of
the so called annihilation and creation (or ``ladder") linear
differential operators $L^-$ and $L^+$ of the first order,
respectively. The corresponding Lie algebra generated by $H$,
$L^-$ and $L^+$ is usually called oscillator or Heisenberg
algebra. Its $m$th-order polynomial generalizations may be then
built from the $(m+1)$th-order linear differential operators $L^-$
and $L^+$, the commutator of which happens to be just an
$m$th-order polynomial in $H$. Such a construction has been known
related to the so called supersymmetric partners of the harmonic
oscillators at even $m$. In the paper it is shown that the same
observation applies also at the odd $m$. In addition, the authors
show that and how the Painlev\'{e} transcendents of types IV and V
emerge in connection with similar algebras (called ``deformed Lie
algebras") at $m=2$ and $m=3$, respectively.