  We show that the umbral correspondence between differential equations can be
achieved by means of a suitable transformation preserving the algebraic
structure of the problems. We present the general properties of these
transformations, derive explicit examples and discuss them in the case of the
App\`{e}l and Sheffer polynomial families. We apply these transformations to
non-linear equations, and discuss how the relevant solutions should be
interpreted.

Exactly Solvable and Integrable Systems