Algebraic invariant curves and first integrals for Riccati polynomial differential systems

© 2014 American Mathematical Society. We study the algebraic invariant curves and first integrals for the Riccati polynomial differential systems of the form x’ = 1, y’ = a(x)y2 + b(x)y+c(x), where a(x), b(x) and c(x) are polynomials. We characterize them when c(x) = κ(b(x) - κa(x)) for some κ ∈ C. We conjecture that algebraic invariant curves and first integrals for these Riccati polynomial differential systems only exist if c(x) = κ(b(x) - κa(x)) for some κ ∈ C.