Peculiar properties of many classical and quantum systems can be related to, or derived from those of a free particle. In this way we explain the appearance and peculiarities of the exotic nonlinear Poincare supersymmetry in reflectionless and finite-gap quantum systems related to the Korteweg-de Vries equation. The same approach is used to explain the origin and the nature of nonlinear symmetries in the perfectly invisible PT-regularized conformal and superconformal mechanics systems.