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We classify non-minimal biconservative surfaces with parallel mean curvaturevector field in $\mathbb{S}^n\times\mathbb{R}$ and$\mathbb{H}^n\times\mathbb{R}$. When these surfaces do not lie in$\mathbb{S}^n$ or $\mathbb{H}^n$ and they are not vertical cylinders, we findtheir explicit (local) equation. We also prove a result on the compactness ofbiconservative surfaces with constant mean curvature in Hadamard manifolds.

CMC biconservative surfaces in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$