Factorization of nonlinear supersymmetry in one-dimensional Quantum Mechanics I: general classification of reducibility and analysis of third-order algebra

Possible factorizations of supersymmetric (SUSY) transformations in one-dimensional Quantum Mechanics into chains of elementary Darboux transformations with non-singular coefficients are investigated. The classification of irreducible (almost) isospectral transformations and of related SUSY algebras is presented. The detailed analysis of SUSY algebras and isospectral operators is done for the third-order case.