I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$ Now clearly this is very difficult, indeed it is equivalent to find the integer solutions of an elliptic curve $E$ defined over $\mathbb{Z}$, in particular $E$ is the smooth plane cubic curve in $\mathbb{P}^2$ given by the equation $$E=(x^3 - 3 x^2 y - 3 x^2 z - 3 x y^2 - 5 x y z - 3 x z^2 + y^3 - 3 y^2 z - 3 y z^2 + z^3=0)$$

Anyway I started playing with this: since $E$ is symmetric with respect to the permutations of the variables $x,y,z$ we have an action $$S_3 \times E \rightarrow E \\ (\sigma,[x,y,z]) \rightarrow [\sigma(x),\sigma(y),\sigma(z)]$$ where I've identified the set $\{x,y,z\}$ with the set $\{1,2,3\}$. We can form the quotient of $E$ with respect to this action yielding a map $$f: E \rightarrow \mathbb{P}^1$$ of degree $6$. By Hurwitz's formula we have that the degree of the ramification divisor $R$ is $12$ and since the ramification points consists in those points with $2$ coordinates equal we have $3$ ramification points of order $4$. My questions now are the following:

- From this description are we able to say that $E$ has infinite integer solutions, maybe noting that this type of elliptic curve is particular?
- It seems to me that $S_3 \subset Aut(E)$ but searching on the internet I found that the automorphism group of elliptic curve can only be of type $\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/6\mathbb{Z}$. What I'm missing?

I'm sorry if I said something wrong, I'm not an expert on this field. Thanks in advance for the help.