On the problem of existence of holomorphic first integrals (Q1910852)

The author considers the system \(x' = B(x,y)\), \(y' = - A(x,y)\), where \(A(x,y)\), \(B(x,y)\) are real analytic functions in a neighbourhood of \(0 \in \mathbb{R}^2\) and \(' = d/dt\), as well as its complexification, i.e. a holomorphic differential equation defined in a neighbourhood of \(0 \in \mathbb{C}^2\), and \(' = d/dz\). To avoid the trivial case the author assumes the \(0 \in \mathbb{R}^2\), and \(0 \in \mathbb{C}^2\) as singular points. With \(\omega (x,y) = A(x,y)dx + B(x,y) dy\) and, a real analytic (resp. holomorphic) first integral of the system \(f : U \to \mathbb{R}^2\), \((f : U \to \mathbb{C}^2)\) and \(df \wedge \omega = 0\), the author proves the following assertion: The problem of the existence of a holomorphic first integral of the given system is algebraically not solvable with respect to the Mattei and Moussu algorithm in a class \(W\) of germs of real analytic 1-form at \(0 \in \mathbb{R}^2\) all having real dynamics of center type, and with first nonvanishing jet of the form \(x^2 + y^2 \{xdx + ydy\}\).