  In this paper, we study biconservative submanifolds in $\mathbb{S}^{n}\times
\mathbb{R}$ and $\mathbb{H}^{n}\times \mathbb{R}$ with parallel mean curvature
vector field and co-dimension 2. We obtain some necessary and sufficient
conditions for such submanifolds to be conservative. In particular, we obtain a
complete classification of 3-dimensional biconservative submanifolds in
$\mathbb{S}^{4}\times \mathbb{R}$ and $\mathbb{H}^{4}\times \mathbb{R}$ with
nonzero parallel mean curvature vector field. We also get some results for
biharmonic submanifolds in $\mathbb{S}^{n}\times \mathbb{R}$ and
$\mathbb{H}^{n}\times \mathbb{R}$.
