Polynomial first integrals of quadratic vector fields

Abstract

We classify all quadratic polynomial differential systems having a polynomial first integral, and provide explicit normal forms for such systems and for their first integrals.

Analytic integrability of quadratic–linear polynomial differential systems

“…In [4] the authors classify the quadratic polynomial differential systems having a polynomial first integral.…”

Quadratic systems with a polynomial first integral: A complete classification in the coefficient space R12

“…But the complete study of all quadratic systems having polynomial first integral was done by Chavarriga, García, Llibre, Pérez del Río and Rodríguez in [7]. They provide the necessary and sufficient conditions in order that a quadratic system has a polynomial first integral.…”

Integrability in nonlinear biomathematical models

“…All quadratic polynomial differential systems having a polynomial first integral where characterized in [4] using normal forms. Moreover in [8] are provided all their phase portraits.…”

“…Moreover in [8] are provided all their phase portraits. In [2] the authors using the results of [4] and applying the invariant theory provide invariant conditions on the coefficients of any non degenerate quadratic system (i.e. quadratic systems having finitely many singular points) in order to determine if it has or not a polynomial first integral without using any normal form.…”