Number of solutions of a system of equation! Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations 
$$
\sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n,
$$
has finite number of solution on $\{\Theta\in{\mathbb T}^n; \theta_1+\ldots+\theta_n =0 \}$.
This equation represents equilibrium point of a dynamical system for $n$ neurons modeled as a network of oscillators. 
Any idea or reference suggestion is appreciated.
 A: Here's an idea. Not a particularly good one, but it's too long for a comment.
Using summation formuli you can rewrite the term in your sum as a polynomial in $\cos(\theta_i)$ and $\sin(\theta_i)$ for $i=1,\ldots,n$. Then you can rationally parametrize your circles by $\cos(\theta_i) =  \frac{1-t_i^2}{1+t_i^2}$, $\sin(\theta_i) =\frac{2t_i}{1+t_i^2}$ and obtain a sum of rational functions thus reducing the problem to a system of polynomial equations. (The last condition $\sum_i \theta_i = 0$ just reduces the number of your variables but you have to pay the price of much uglier expressions coming from $\sin$ containing say $\theta_n = -\sum_{i\neq n} \theta_i$.)
For fixed $n$ you may (hopefully) deduce finiteness of the solution set via Groebner basis.
Or maybe there's even better coordinate system for the cut of your torus (which is, if I am not mistaken, topologically just the torus of dimension one less $\simeq \mathbb{T}^{n-1}$) and the functions in your equations have some "geometrical meaning".
