Show that we can only find $6$ biholomorphic mappings $\phi$.

Question

I have a problem:

Let $U$ be a connected open subset of a point $(0,0)$, and $$ M= \{(z,\ w) \in \Bbb C^2: \text{Im}w = |z w|^2+|z|^8+\dfrac{15}{7} |z|^2 \text{Re}z^6 \}$$ Let $$\phi:\ U \to \Bbb C^2$$ such that $$\phi (0,0)=(0,0),\ \ \phi (M)\subset M \tag{*}$$

Show that we can only find $6$ biholomorphic mappings $\phi$ such that $(*)$.

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Any help will be appreciated! Thanks!

Answer 1

How about

$$\phi_k(z,w) = (ze^{2\pi i k/6}, w)$$

for $k=0,\ldots 5$. Note that $w$, $|z|$ and $z^6$ are invariant under $\phi_k$, and hence so is $M$.