I am trying to find complex solutions with positive real part $\{t_j \;|\;{\rm Re}\;t_j>0, j = 1, 2, 3, \dots, n\}$ of the system of equations $$0 = 1 + \sum_j \left(t_j^{2l+1} + {t_j^*}^{2l+1}\right),\; l = 1,2,3,\dots m.$$ Where for a given $n$ I would like to make $m$ as large as possible. Since, this system is non-analytic and thus for $n=m$ most likely under-constrained, I had the idea to just fix the magnitude of all solutions to 1: $t_j = e^{i\phi_j}$ with $-{\pi\over 2} < \phi_1 \le \phi_2 \le \dots \le \phi_n <{\pi\over 2}$. In terms of these the system becomes: $$0 = 1 + 2\sum_j \cos{\left[\phi_j(2l+1)\right]},\; l = 1,2,3,\dots n.$$ This definitely has solutions up to $n=m=2$, but already for $n=3$, my naive attempt at numerically solving this (Mathematica's NSolve) is taking quite long. Is there some better way to find or at least confirm the existence of such solutions?

Thanks, Nik