Exact solutions of the Cauchy problem for the Friedman equation

We reduce the cosmological Friedmann equation for a universe filled with a scalar field to a system of two first-order equations, one of which is an equation with separable variables. For this equation, we write exact solutions for a quadratic potential as a series in the helical and attractor domains and also for quite arbitrary potentials both in the neighborhood of a finite point and in a neighborhood of infinity. We prove the existence and uniqueness of classical solutions of the Cauchy problem of the Friedmann equation in certain cases and the presence of exactly two solutions in other cases.