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We consider the existence of a Noether symmetry in the
scalar-tensor theory of gravity in flat Friedman-Robertson-Walker
(FRW) cosmology. The forms of coupling function $\omega(\phi)$ and
generic potential $V(\phi)$ are obtained by requiring the
existence of a Noether symmetry for such theory. We derive exact
cosmological solutions of the field equations from a point-like
Lagrangian.
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