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Global properties of a class of planar vector fields of infinite type. (English) Zbl 0882.35029
Consider a 1-form $$\omega= A(x,y)dx+ B(x,y)dy$$ in $$\mathbb{R}^2$$ with no singularities, that is, $$\omega(p)\neq 0$$ for every $$p$$. We assume that $$\omega$$ is rotationally invariant. The orthogonal to $$\omega$$ is generated by the vector field $L=- B(x,y) {\partial\over\partial x}+ A(x,y) {\partial\over\partial y}.$ We say $$L$$ (or $$\omega$$) has a first integral in an open set $$\Omega$$ if there is a differentiable function $$Z:\Omega\to\mathbb{C}$$ such that $$LZ=0$$ in $$\Omega$$ and $$dZ(p)\neq 0$$ for every $$p$$ in $$\Omega$$. The existence of such functions is crucial for the study of local and global properties of the vector field $$L$$ such as solvability, regularity, unique continuation, Liouville’s property, etc. When $$\omega$$ is assumed to be $$C^\infty$$ and complex-valued, we showed in [Manuscr. Math. 89, 355-371 (1996; Zbl 0858.35021)] that it has a global first integral in $$\mathbb{R}^2$$ whenever it is of finite type. The first integral was then used to describe several properties of $$L$$.
As a natural sequel, here we study $$\mathbb{C}$$-valued, real analytic forms $$\omega$$ that are of infinite type along certain of its characteristic circles. In polar coordinates, such a structure is uniquely written as $$M(r)dr+ id\theta$$, where $$M$$ is a meromorphic function of the variable $$r$$. The structure is of infinite type along the circle with radius $$\rho$$ if $$M(\rho)=\infty$$. It is shown that the residue of $$M$$ at $$\rho$$ is an invariant of the structure generated by $$\omega$$ and that it plays a fundamental role in the integrability and solvability questions.

##### MSC:
 35F05 Linear first-order PDEs
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##### References:
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