Flat space isotropic cosmological models including a nonminimally coupled scalar field with a nonlinear potential are studied within the Einstein-Kartan theory. The exact general solutions for various types of scalar fields are derived for arbitrary values of the coupling constant ξ. It is shown that both singular and nonsingular models are feasible. The spectral values of ξ and restriction on ξ are found for the foregoing solutions. The cosmological consequence of taking into account the scalar-field potential is discussed.