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 $16$).
Categories: math.AG math.DG