Preprint : Paderborn April, 1991 Nilpotent and Recursive Flows Benno Fuchssteiner University of Paderborn D 4790 Paderborn Germany Mauro Lo Schiavo Dipartimento di M.M.M. per le Sc. Appl. Universit` a La Sapienza I 00161 Roma In this paper we introduce a class of nonlinear vector fields on infinite di- mensional manifolds such that the corresponding evolution equations can be solved with the same method one uses to solve ordinary differential equa- tions with constant coefficients. Mostly, these equations are nonlinear par- tial differential equations. It is shown that these flows are characterized by a generalization of the ’method of variation of constants’ which is widely used for second order problems to find general solutions out of particular ones. Invariant densities are constructed for these flows in a natural way. These invariant densities are providing an essential tool for solving initial value and boundary value problems for the equations under consideration. Many applications are presented 1 Introduction In this section we give the essential definitions and illustrate them by a num- ber of examples. Definition 1.1: We consider a manifold given by some vector space E, and we denote by v the typical element of E. A vector field G(v) on E is said to be be • nilpotent if, whenever the equation s t = G(s) defines a flow (s, t) → s(t) , then this flow satisfies ( d dt ) N+1 s =0 (1.1) 1 Manuscripta Mathematica, 79, p.27-48, 1993