Invariant constant mean curvature tubes around a horizontal geodesic in $\mathbb{E}(\kappa,\tau)$-spaces

We consider constant mean curvature surfaces (invariant by a continuous group of isometries) lying at bounded distance from a horizontal geodesic on any homogeneous $3$-manifold $\mathbb{E}(\kappa,\tau)$ with isometry group of dimension $4$. These surfaces are called horizontal tubes. We show that they foliate $\mathbb{E}(\kappa,\tau)$ minus one or two horizontal geodesics provided that $(1-x_0^2)\kappa+4\tau^2\leq 0$, where $x_0\approx 0.833557$. We also describe precisely how horizontal and vertical geodesics get deformed by Daniel's sister correspondence and conclude that the family of horizontal tubes is preserved by the correspondence. These tubes are topologically tori in $\mathbb{S}^2\times\mathbb{R}$ and Berger spheres, in which case we compute their conformal type and analyze numerically their isoperimetric profiles.