  Let $M$ be a simply connected homogeneous three-manifold with isometry group
of dimension $4$, and let $\Sigma$ be any compact surface of genus zero
immersed in $M$ whose mean, extrinsic and Gauss curvatures satisfy a smooth
elliptic relation $\Phi(H,K_e,K)=0$. In this paper we prove that $\Sigma$ is a
sphere of revolution, provided that the unique inextendible rotational surface
$S$ in $M$ that satisfies this equation and touches its rotation axis
orthogonally has bounded second fundamental form. In particular, we prove that:
(i) any elliptic Weingarten sphere immersed in $\mathbb{H}^2\times \mathbb{R}$
is a rotational sphere. (ii) Any sphere of constant positive extrinsic
curvature immersed in $M$ is a rotational sphere, and (iii) Any immersed sphere
in $M$ that satisfies an elliptic Weingarten equation $H=\phi(H^2-K_e)\geq a>0$
with $\phi$ bounded, is a rotational sphere. As a very particular case of this
last result, we recover the Abresch-Rosenberg classification of constant mean
curvature spheres in $M$.
