Projective superflows. III. Finite subgroups of $U(2)$

Let $X\in {\mathbb{R}}^n$ or ${\mathbb{C}}^n$. For $\varphi :{\mathbb{R}}^n\mapsto {\mathbb{R}}^n$ (respectively, $\varphi :{\mathbb{C}}^n\mapsto {\mathbb{C}}^n$) and $t\in \mathbb{R}$ (respectively, $\mathbb{C}$), we put ${\varphi }^t=t^{-1}\varphi (Xt)$. A projective flow is a solution to the projective translation equation ${\varphi }^{t+s}={\varphi }^t\circ {\varphi }^s$, $t,s\in \mathbb{R}$ or $\mathbb{C}$. The projective superflow is a projective flow with a rational vector field which, among projective flows with a given symmetry, is, up to a homothety, unique and optimal. In the first and the second part of this work we classified real $2$ and $3-$dimensional supeflows over $\mathbb{R}$. In this third part we classify all $2-$dimensional complex superflows; that is, whose group of symmetries are finite subgroups of $U(2)$. This includes both irreducible and reducible superflows.