Galileon hairs of Dyson spheres, Vainshtein's coiffure and hirsute bubbles

We study the fields of spherically symmetric thin shell sources, a.k.a. Dyson spheres, in a fully nonlinear covariant theory of gravity with the simplest galileon field. We integrate exactly all the field equations once, reducing them to first order nonlinear equations. For the simplest galileon, static solutions come on six distinct branches. On one, a Dyson sphere surrounds itself with a galileon hair, which far away looks like a hair of any Brans-Dicke field. The hair changes below the Vainshtein scale, where the extra galileon terms dominate the minimal gradients of the field. Their hair looks more like a fuzz, because the galileon terms are suppressed by the derivative of the volume determinant. It shuts off the 'hair bunching' over the 'angular' 2-sphere. Hence the fuzz remains dilute even close to the source. This is really why the Vainshtein's suppression of the modifications of gravity works close to the source. On the other five branches, the static solutions are all singular far from the source, and shuttered off from asymptotic infinity. One of them, however, is really the self-accelerating branch, and the singularity is removed by turning on time dependence. We give examples of regulated solutions, where the Dyson sphere explodes outward, and its self-accelerating side is nonsingular. These constructions may open channels for nonperturbative transitions between branches, which need to be addressed further to determine phenomenological viability of multi-branch gravities.