Preprint : Paderborn October, 1991 A new class of nonlinear partial differential equations solvable by quadratures 1 Benno Fuchssteiner University of Paderborn D 4790 Paderborn Germany Sandra Carillo Dipartimento di M.M.M. per le Sc. Appl. Universit` a ”La Sapienza” I 00161 Roma Abstract We introduce a class of nonlinear partial differential equations in two independent scalar variables, say x and t, characterized by the property that the initial value problem for given boundary values can be solved by quadratures. The Liouville equation enjoys such a prop- erty and seems to be the most simple equation among the elements of this class. Hence we term these equations generalized Liouville equa- tions. We further introduce the Riccati property, which refers to nonlinear ordinary differential equations and generalizes a well known property of the Riccati equation. This property requires that, whenever one par- ticular solution of an equation is given, then it is possible to construct from that the general solution by quadratures. Nonlinear ordinary dif- ferential equations which enjoy the Riccati property are shown to be related to generalized Liouville equations. 1 work partially supported by the G.N.F.M. of C.N.R. and by the M.U.R.S.T. project Geometria e Fisica. One of the authors (B. F.) wishes to acknowledge his gratitude to the Universit`a ”La Sapienza” and the Italian C.N.R. for the kind hospitality he received in Roma. 1 in: Geometry and Analysis: Trends in Teaching and Research B. Fuchssteiner and W. A. J. Luxemburg, eds. Bibliographisches Institut Mannheim, p.73-85, 1992