Invent. math. 93, 545-555 (1988) mathematicae 9 Springer-Vedag 1988 On codimension one submersions of euclidean spaces Antonio F. Costa 1, F.G. Gascon 2, ,, and A. Gonzalez-Lopez 3 Departomento de Matematicas Fundamentales, UNED, 28040 Madrid, Spain z Departomento de M&odos Matem/tticos, Facultad de C. Fisicas, Universidad Complutense, 28040 Madrid, Spain 3 School of Mathematics, Minnesota University, Minneapolis, MN 55455, USA Summary. It is shown that when (n- 1) first integrals of a dynamical system X in R" are known (and they are independent at any point of R") then one can have (n > 3) that certain orbits of X are diffeomorphic to circles and others diffeomorphic to straight-lines. An analytical criterion is also given (involving only the first derivatives of the first integrals) in order that all the orbits be diffeomorphic to straight lines. Therefore the criterion is sufficient in order to avoid the presence of geometrical chaos among the orbits of the dynamical system X. I. Introduction In a recent paper on Szebehely's equation I-3] the following question was raised: Given the autonomous dynamical system xeR n j, (I.1) and assuming that (n-1) functionally independent and smooth first integrals f~of (I.1) are known, that is rank (Df)v = n- 1"[ (I.2) for every P of RnJ ' Where f is the mapping of R" in R"-1 whose components are fl ..... f.-1, then it is well known that the connected components (orbits) of the sets f-1 (a) (a~R"-1) are either topological circles or topological straight lines [4, 5]. That ~s, the condition rank f= n-1 for every point of R" implies that orbits can be either circles or straight lines; is it possible to have some orbits of type "circles" and others of type "straight lines" and still keeping the condition * Author to which all the correspondence should be addressed