Cosmological Solutions of Integrable <i>F</i>(<i>R</i>) Gravity Models with an Additional Scalar Field

We consider <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> cosmological models with a scalar field. For the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>R</mi><mn>2</mn></msup></semantics></math></inline-formula> model in the spatially flat Friedmann–Lemaître–Robertson–Walker metric, the Ricci scalar <i>R</i> can smoothly change its sign during the evolution if and only if the scalar field is a phantom one. In the Bianchi I metric, the Ricci scalar cannot smoothly change its sign if the corresponding solution is anisotropic at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. This result does not depend on the type of the scalar field. In the Bianchi I metric, the general solution of evolution equations has been obtained.