Biconservative surfaces in ${\mathbb{S}}^2\times \mathbb{R}$ and ${\mathbb{H}}^2\times \mathbb{R}$

We find the explicit local equations of biconservative surfaces with non-constant mean curvature in ${\mathbb{S}}^2\times \mathbb{R}$ and ${\mathbb{H}}^2\times \mathbb{R}$, when the gradient of the mean curvature function is a principal direction.