Local analytic integrability for a class of degenerate planar vector fields

Abstract Using the normal form theory and the existence of an algebraic inverse integrating factor we characterize the local analytic integrability of the systems whose quasi-homogeneous leading term is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>5</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfenced> </m:math> $\left({a}_{1}{y}^{3}+{a}_{2}{x}^{3}y,{b}_{1}{x}^{5}+{b}_{2}{x}^{2}{y}^{2}\right)$ . More specifically we prove that the analytic integrable vector fields inside such family are orbitally equivalent to a semi-quasi-homogeneus system, that is, are not orbitally equivalent to its lowest-degree quasi-homogeneous term.