SUSY partners of the truncated oscillator, Painlev\'e transcendents and B\"acklund transformations

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.