@NoamD.Elkies Besides your observation, there is also a change of variables of the form $X_r=\pm\zeta^rY_r$ to kill the powers of -1 in the first two lines simultaneously; here $\zeta$ is a $m/2$ or $2m$-th root of unity according to $4$ divides $m$ or not.

@NoamD.Elkies I came across this system in a study for power residue difference set. I found that if the subgroup of index $m$ in $\mathbb{F}_p^\times$ form a difference set of $\mathbb{F}_p^+$, then $X_r\equiv\Gamma_p(\frac{r}{m})\equiv\frac{1}{(r(p-1)/m)!}\pmod{p}$ should satisty the first line of equations. The last two lines of equations come from the own properties of $p$-adic Gamma functions (here I'm considering the case when $2$ is a $m$-th power in $\mathbb{F}_p$). My origional purpose is to exclude the existence of certain power residue difference sets by showing it has no solutions.

@PadraigÓCatháin Yes, $n$ means the order $k-\lambda$. Do you mean given any $v$, $N_v$ should be equal to $N_{v,n}$ for some $n$? And what is the title of Muzychuk's paper you mentioned?:)

For a given $m$, provided we have know that the system has no solution over $\mathbb{C}$, this paper says that the number of primes $p$ such that the system has solution over $\mathbb{F}_p$ is bounded. But does this provide a concrete bound $N$ such that the system has no solution whenever $p>N$?

@B.Wellington Thanks to your comment. I've changed $2X_{2t}$ to $2X_{2s}$ now.

@B.Wellington The first equation is $2X_2-X_1^2=0$, so $X_2=\frac{X_1^2}{2}$. In fact, the first line of equations shows that $X_{2t}$ is a polynomial of $X_1,\dots,X_{2t-1}$ for $1\le t\le\frac{m}{2}-1$.