Geodesics and shortest arcs of some sub-Riemannian metrics on the Lie groups $\operatorname{SU}(1,1)\times\Bbb{R}$ and $\operatorname{SO}_0(2,1)\times\Bbb{R}$ with three-dimensional generating distributions

We find geodesics, shortest arcs, cut loci, and first conjugate loci for some left-invariant sub-Riemannian metrics on the Lie groups $\operatorname{SU}(1,1)\times\Bbb{R}$ and $\operatorname{SO}_0(2,1)\times\Bbb{R}$.