On the polynomial first integrals of certain second-order differential equations

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Abstract
It is shown that any first integral of type P2(ẋ)—a polynomial of degree 2 in ẋ—of the differential equation ẍ=Vx can be obtained from a pointlike gauge symmetry of the action AL associated to L= 1/2 ẋ2+V(t,x). The same result holds for any first integral of kind Pn(ẋ) when dynamical symmetries of AL polynomials in ẋ are allowed. The neccessary and sufficient conditions that V(t,x) must satisfy in order that ẍ=Vx possesses a first integral of type Pn(ẋ) have been obtained. These conditions reduce (when n=2) to a condition obtained by Leach. The computational advantages and difficulties which appear in order to obtain first integrals for type Pn(ẋ) are also briefly discussed.

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