for which values of $\theta$.does this equation: $x_1^{\sin\theta}+\cdots+x_n^{\sin\theta}=1$ have rational solutions for all $n$?

Question

I'd surprised if this equation:$$x_1^{\sin\theta}+\cdots+x_n^{\sin\theta}=1 $$ have rational solutions for all $n$ and for all values of $\theta$.

My question here is: for which values of $\theta$ does this equation: $$x_1^{\sin\theta}+\cdots+x_n^{\sin\theta}=1$$ have rational solutions for all $n$ ?

Thank you for any kind of help .

Answer 1

How about $x_1 = 1$, $x_2 = \ldots = x_n = 0$, as long as $\sin\theta > 0$ (I won't venture an opinion on the case $\sin\theta = 0$).