  In this work the supersymmetric technique is applied to the truncated
oscillator to generate Hamiltonians ruled by second and third-order polynomial
Heisenberg algebras, which are connected to the Painlev\'e IV and Painlev\'e V
equations respectively. The aforementioned connection is exploited to produce
particular solutions to both non-linear differential equations and the
B\"acklund transformations relating them.
