Projective superflows. II. $O(3)$ and the icosahedral group

Abstract

Let $X\in\mathbb{R}^{n}$. For $\phi:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $\phi^{t}=t^{-1}\phi(Xt)$. A projective flow is a solution to the projective translation equation $\phi^{t+s}=\phi^{t}\circ\phi^{s}$, $t,s\in\mathbb{R}$. The projective superflow is a projective flow with a rational vector field which, among projective flows with a given symmetry, is in a sense unique and optimal. In this second part we classify $3$-dimensional real superflows. Apart from the superflow $\phi_{\hat{\mathbb{T}}}$ (with a group of symmetries being all symmetries of a tetrahedron) and the superflow $\phi_{\mathbb{O}}$ (with a group of symmetries being orientation preserving symmetries of an octahedron), both described in the first part of this study, here we investigate in detail the superflow $\phi_{\mathbb{I}}$ whose group of symmetries is the icosahedral group $\mathbb{I}$ of order $60$. This superflow is a flow on co-centric spheres, and is also solenoidal. These three superflows is the full (up to linear conjugation) list of $3$-dimensional irreducible real projective superflows. We also find all reducible $3$-dimensional real superflows. There are two of them: one with group of symmetries being all symmetries of a $3$-prism (group of order $12$), and the second with a group of symmetries being all symmetries of a $4$-antiprism (group of order $16$).