  In this article, we study constant mean curvature isometric immersions into
$\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we
classify these isometric immersions when the surface has constant intrinsic
curvature. As applications, we use the sister surface correspondence to
classify the constant mean curvature surfaces with constant intrinsic curvature
in the $3-$dimensional homogenous manifolds $\mathbb{E}(\kappa, \tau)$ and we
use the Torralbo-Urbano correspondence to classify the parallel mean curvature
surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{H}^2 \times
\mathbb{H}^2$ with constant intrinsic curvature. It is worthwhile to point out
that these classifications provide new examples.
