We consider modified gravity cosmological models that can be transformed into two-field chiral cosmological models by the conformal metric transformation. For the $R^2$ gravity model with an additional scalar field and the corresponding two-field chiral cosmological model with the cosmological constant and nonstandard kinetic part of the action, the general solutions have been obtained. We analyze the correspondence of the cosmic time solutions obtained and different possible evolutions of the Hubble parameters in the Einstein and Jordan frames. The talk is based on the paper V.R. Ivanov, S.Yu. Vernov, Eur. Phys. J. C 81 (2021) 985 [https://doi.org/10.1140/epjc/s10052-021-09792-4] and recent investigations.