General solutions for a flat Friedmann universe filled with a perfect fluid and a scalar field with an exponential potential

Summary: "We study the integrability by quadratures of a spatially flat Friedmann model containing both a minimally coupled scalar field phi with an exponential potential V(phi)simexp[-sqrt 6sigmakappaphi], kappa=sqrt{8pi G_N}, of arbitrary sign and a perfect fluid with the barotropic equation of state p=(1-h)rho. From a mathematical viewpoint, the model is a pseudo-Euclidean Toda-like system with 2 degrees of freedom. We apply the methods developed in our previous papers, based on the Minkowski-like geometry for 2 characteristic vectors depending on the parameters sigma and h. In the general case, the problem is reduced to the integrability of a second-order ordinary differential equation known as the generalized Emden-Fowler equation, which was investigated by discrete-group methods. We present 4 classes of general solutions for the parameters obeying the following relations: (A) sigma is arbitrary, h=0; (B) sigma=1-h/2, 0