On non-integrability of general systems of differential equations

A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals

“…We shall use the following result known by Poincaré [16]: If a polynomial system (1) has a rational first integral, then the eigenvalues λ 1 and λ 2 associated to any singular point of the system must be resonant in the following sense: there exist nonnegative integers m 1 and m 2 with m 1 + m 2 ≥ 1 such that m 1 λ 1 + m 2 λ 2 = 0. For a proof see, for example, [9] or [11]. Now it is easy to check that the origin of system (2) is a singular point whose ratio of eigenvalues is 1 − c. Since c is irrational, the system can have no rational first integral.…”

Liouvillian and Analytic First Integrals for the Brusselator System

“…To do so, we need the following auxiliary result, it is due to Poincaré in [19], see also [6] for a direct proof. Through the paper Z + will denote the set of non-negative integers.…”

A Necessary Condition for Existence of Lie Symmetries in General Systems of Ordinary Differential Equations

“…Recently Furta [5] and Goriely [6] independently overcame this weak point although the assertion is a little weak. Because this method does not suppose non-degeneracy of first integrals, one can use the method for proofs of nonexistence of first integrals.…”

“…Definition 7 (Semi-quasihomogeneous systems [5]). The system (27) is called a semi-quasihomogeneous system if it can be expressed as the forṁ…”