Ricci Soliton Classification on $\mathbb H^2\times\mathbb R$

We study Ricci solitons on the Riemannian manifold $\mathbb H^2\times\mathbb R$ equipped with the standard metric. A complete classification of soliton vector fields is obtained: they form a four-dimensional affine space, namely a translate of the Killing algebra $\mathfrak{isom}(\mathbb H^2\times\mathbb R)$. All corresponding solitons are expanding. In addition, gradient solitons are fully characterized and shown to form a one-parameter subfamily of the complete family of soliton fields. As a byproduct, every soliton vector field turns out to be affine, preserving the Levi-Civita connection, the curvature tensor, and the Ricci tensor.