  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 model with the cosmological constant and nonstandard
kinetic part of the action, the general solutions have been obtained in the
spatially flat FLRW metric. We analyze the correspondence of the cosmic time
solutions obtained and different possible evolutions of the Hubble parameters
in the Einstein and Jordan frames.
