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We study the fields of spherically symmetric thin shell sources, a.k.a. Dysonspheres, in a {\it fully nonlinear covariant} theory of gravity with thesimplest 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 {\it six} distinct branches. On one, a Dyson spheresurrounds itself with a galileon hair, which far away looks like a hair of anyBrans-Dicke field. The hair changes below the Vainshtein scale, where the extragalileon terms dominate the minimal gradients of the field. Their hair looksmore like a fuzz, because the galileon terms are suppressed by the derivativeof 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 isreally why the Vainshtein's suppression of the modifications of gravity worksclose to the source. On the other five branches, the static solutions are all{\it singular} far from the source, and shuttered off from asymptotic infinity.One of them, however, is really the self-accelerating branch, and thesingularity is removed by turning on time dependence. We give examples ofregulated solutions, where the Dyson sphere explodes outward, and itsself-accelerating side is nonsingular. These constructions may open channelsfor nonperturbative transitions between branches, which need to be addressedfurther to determine phenomenological viability of multi-branch gravities.

Galileon Hairs of Dyson Spheres, Vainshtein's Coiffure and Hirsute Bubbles