  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
$\phi^{t+s}=\phi^{t}\circ\phi^{s}$, $t,s\in\mathbb{R}$.
  Previously we have developed an arithmetic, topologic and analytic theory of
$2$-dimensional projective flows: rational, algebraic, unramified, abelian
flows, commuting flows. The current paper is devoted to highly symmetric flows
- superflows. Within flows with a given symmetry, superflows are unique and
optimal.
  Our first result classifies all $2$-dimensional superflows. For any positive
integer $d$, there exists the superflow $\phi_{\mathbb{D}_{2d+1}}$ whose group
of symmetries is the dihedral group $\mathbb{D}_{2d+1}$. In the current paper
we explore the superflow $\phi_{\mathbb{D}_{5}}$, which leads to investigation
of abelian functions over curve of genus $6$.
  The $3$-dimensional theory of projective flows is more involved. We
investigate two different $3$-dimensional superflows, whose group of symmetries
are, respectively, the full tetrahedral group $\widehat{\mathbb{T}}$ (all
symmetries of a tetrahedron), and the octahedral group $\mathbb{O}$
(orientation preserving symmetries of an octahedron), both isomorphic, though
non-equivalent as representations. The generic orbits of the first flow are
space curves of genus $1$, and the flow itself can be analytically described in
terms of Jacobi elliptic functions. The generic orbits of the second flow are
curves of genus $9$, and the flow itself can be described in terms of
Weierstrass elliptic functions (via reduction of hyper-elliptic functions to
elliptic).
  In the second part of this work we will classify all $3$-dimensional
superflows (including the icosahedral superflow), and in the third we
investigate superflows over $\mathbb{C}$.
