  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 Poincar\'e 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 $\mathcal{PT}$-regularized
conformal and superconformal mechanics systems.
