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We present the solution space for the case of a minimally coupled scalarfield with arbitrary potential in a FLRW metric. This is made possible due tothe existence of a nonlocal integral of motion corresponding to the conformalKilling field of the two-dimensional minisuperspace metric. The case for bothspatially flat and non flat are studied first in the presence of only thescalar field and subsequently with the addition of non interacting perfectfluids. It is verified that this addition does not change the general form ofthe solution, but only the particular expressions of the scalar field and thepotential. The results are applied in the case of parametric dark energy modelswhere we derive the scalar field equivalence solution for some proposed modelsin the literature.

General Analytic Solutions of Scalar Field Cosmology with Arbitrary Potential