The missing first integral
The author studies the question of integrability of abstract CR structures on \({\mathbb{R}}^ 3\). If a is a smooth function defined near the origin in \({\mathbb{R}}^ 3\), he says that a satisfies condition \((I^+)\) if there exists a neighborhood of the origin U in \({\mathbb{R}}^ 4\) and \(A\in C^{\infty}(U)\) such that \(\partial A/\partial t+i\partial A/\partial s=0\) in \(U^+=\{(x,y,t,s)\in U :\) \(s>0\}\) and \(A(x,y,t,0)=a(x,y,t)\) for \((x,y,t)\in {\mathbb{R}}^ 3\) with (x,y,t,0)\(\in U\). He proves the following results: (1) If a structure is integrable, then there exist coordinates (x,y,t) near the origin in \({\mathbb{R}}^ 3\) such that in these coordinates we have (modulo a smooth nonvanishing factor) \(L=\partial /\partial \bar z+a \partial /\partial t\) where the function a satisfies condition \((I^+)\). (2) If \(L=\partial /\partial \bar z+a \partial /\partial t\) where \(a(0)=0\), \(Im(\partial a/\partial z)(0)<0\) and condition \((I^+)\) holds, then L is integrable.
scientific article; zbMATH DE number 4146511
scientific article