I want to find a solution to

$$ \frac{x\_1}{x\_2 + x\_3} +\frac{x\_2}{x\_1 + x\_3} + \frac{x\_3}{x\_1+x\_2} = 4 $$

for $x\_1,x\_2,x\_3>0$, and $x\_1+x\_2+x\_3=1$.

We have
$$\frac{x\_1}{1-x\_1}+\frac{x\_2}{1-x\_2}+\frac{1-x\_1-x\_2}{x\_1+x\_2}=4.$$

Let $\dfrac{x\_1}{1-x\_1}=s,\dfrac{x\_2}{1-x\_2}=t,$then $x\_1=\dfrac{s}{1+s},x\_2=\dfrac{t}{1+t}.$
We then have $$s+t+\dfrac{1-st}{s+t+2st}=4,$$
and $s=x+y,t=x-y,$then$$\frac{4 x^3+3 x^2-4 x y^2+y^2+1}{2 \left(x^2+x-y^2\right)}=4$$
$$y^2=\frac{4 x^3-5 x^2-8 x+1}{4 x-9}$$
which eventually simplifies to solving $$y^2 = x^3 + 121 x^2 + 1144 x + 2704.$$
How can we solve this? I am unfamiliar with eliptic curves, even after reading some basic material, it seems each equation is tackled differently. Is there a routine method for equations like this?