ALGEBRAIC DEGENERACY AND PARTIAL INTEGRABILITY FOR SYSTEMS OF ORDINARY DIFFERENTIAL-EQUATIONS

We show that a certain type of representation of ordinary differential equations can be used together with nonlinear transformations to find integrability conditions and to construct the corresponding first integrals. These conditions are obtained through the study of the rank of one of the matrices introduced by the representation. We also point out that the existence of particular time-dependent first integrals is bijectively connected to the algebraic degeneracy of this matrix. Some examples connected with nonlinear sciences are analyzed and the comparison with other methods is presented. © 1990.