  In this work, some new integrable and nonintegrable cosmological models of
the Ho$\check{r}$ava-Lifshitz gravity are proposed. For some of them, exact
solutions are presented. Then these results extend for the F(R)
Ho$\check{r}$ava-Lifshitz gravity theory case. In particular, several
integrable cosmological models of this modified gravity theory were constructed
in the explicit form.
