LAUSR

The Klein four-group symmetry of the eigenvalue problem equation for the conformal mechanics model of de Alfaro-Fubini-Furlan (AFF) with coupling constant $g=\nu(\nu+1)\geq -1/4$ undergoes a complete or partial (in the case of half-integer values of $\nu$) breaking at the level of eigenstates of the system. We exploit this breaking of discrete symmetry to construct the dual Darboux transformations which generate the same but spectrally shifted pairs of rationally deformed AFF models for any value of the parameter $\nu$. Two distinct pairs of intertwining operators associated with Darboux duality allow us to construct the complete sets of spectrum generating ladder operators which detect and describe finite-gap structure of each deformed system and generate three distinct species of nonlinearly deformed $\mathfrak{sl}(2,{\mathbb R})$ algebra that give rise to nonlinear deformations of the extended super-conformal structure. We show that at half-integer values of $\nu$, the Jordan states associated with confluent Darboux transformations enter the construction, and the spectrum of rationally deformed AFF systems suffers structural changes.