Abstract
A string-inspired effective theory of gravity, containing Gauss-Bonnet invariant interacting with a scalar field, is considered in view of obtaining cosmological dark energy solutions. A Lagrange multiplier is inserted into the action in order to achieve the cosmological reconstruction by selecting suitable forms of couplings and potentials. Several cosmological exact solutions (including dark energy of quintessence, phantom, or Little Rip type) are derived in the presence and in the absence of the Lagrange multiplier showing the difference in the two dynamical approaches. In the models that we consider, the Lagrange multiplier behaves as a sort of dust fluid that realizes the transitions between matter-dominated and dark energy epochs. The relation between Lagrange multipliers and Noether symmetries is discussed.