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35-XX Partial differential equations
35Nxx Overdetermined systems
35N10 Overdetermined systems with variable coefficients

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$L^{2}$ -estimates and existence theorems for generalized analytic functions in several variables.

Medzhidov, Z.G. (2004)

Sibirskij Matematicheskij Zhurnal

Liouvillian first integrals of homogeneouspolynomial 3-dimensional vector fields

Jean Moulin Ollagnier (1996)

Colloquium Mathematicae

Given a 3-dimensional vector field V with coordinates $V_{x}$ , $V_{y}$ and $V_{z}$ that are homogeneous polynomials in the ring k[x,y,z], we give a necessary and sufficient condition for the existence of a Liouvillian first integral of V which is homogeneous of degree 0. This condition is the existence of some 1-forms with coordinates in the ring k[x,y,z] enjoying precise properties; in particular, they have to be integrable in the sense of Pfaff and orthogonal to the vector field V. Thus, our theorem links the existence...

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