Invariants of homogeneous ordinary differential equations (Q1290240)

The class of smooth vector fields \(X\) in an open subset \(M\subset \mathbb{R}^n\) which are homogeneous with respect to a fixed vector field \(Y\) and the corresponding ordinary differential equations are considered. The local existence of \(n-1\) \(Y\)-homogeneous first integrals near the nondegenerate with respect to both (\(X\) and \(Y\)) vector fields point \(x_0\in M\) is proved. The problem of a reduction of the \(Y\)-homogeneous \(n\)-dimensional system to an \(n-1\)-dimensional one and relationships between the first integrals of the initial and reduced systems are studied. The problem of the local existence of smooth invariant measures for the \(Y\)-homogeneous systems is analyzed. Applications of this theory to some special classes of \(3\)-dimensional systems are considered.