On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation

We demonstrate the way to derive second Painlevé equation P2 and its Bäcklund transformations from deformations of Nonlinear Schrödinger (NLS), all while preserving strict invariance with respect Schlesinger transformations. The proposed algorithm allows for a construction Jordan algebra-based completely integrable multiple-field generalizations also producing corresponding suggest calling such models JP-systems. For example, algebra JMat(N,N) product in form semi-anticommutator is shown generate an matrix generalization P2, whereas VN produces different JP-system that serves as Sokolov’s vectorial NLS.