New approach to find exact solutions for cosmological models with a scalar field

We study the cosmological models whose ordinary differential equations can be given a Lagrangian description on a two-dimensional "configuration space." By requiring the existence of a N\"other symmetry for such a Lagrangian, we are able to show that the potential must have an expotential form. With the help of the constants of motion we can get from that Lagrangian, we integrate the models and analyze the behavior for all the possible varying free parameters and plot some of the solutions. We also compare our results with others available in the literature.