Eisenhart-Duval lift, Nonlocal Conservation laws and Painlevé Analysis in Scalar field Cosmology

We investigate the existence of nonlocal conservation laws for the gravitational field equations of scalar field cosmology in an FLRW background with a dust fluid source. We perform such analysis by using a novel approach for the Eisenhart-Duval lift. It follows that the scalar field potential V( ϕ) =α( eλϕ+β) admits nontrivial conservation laws. Furthermore, we employ the Painlevé analysis to examine the integrability of the field equations. For the quintessence model, we establish that the cosmological field equations possess the Painlevé property and are integrable for λ²>6. In contrast, for the phantom scalar field, the cosmological field equations exhibit the Painlevé property for any value of the parameter $% λ$. We present analytic solutions expressed in terms of Right Laurent expansions for various values of the parameter $λ$. Finally, we discuss the qualitative evolution of the effective equation of state parameter for these analytic solutions.