  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 presence and in 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.
