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

Abstract

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.