On determination of Zero-sum $\ell$-generalized Schur Numbers for some linear equations

Let $r$, $m$ and $k\geq 2$ be positive integers such that $r\mid k$ and let $v \in \left[ 0,\lfloor \frac{k-1}{2r} \rfloor \right]$ be any integer. For any integer $\ell \in [1, k]$ and $\epsilon \in \{0,1\}$, we let $\mathcal{E}_{v}^{(\ell, \epsilon)}$ be the linear homogeneous equation defined by $\mathcal{E}_{v}^{(\ell, \epsilon)}: x_1 + \cdots + x_{k-(rv+\epsilon)} =x_{k-(rv+\epsilon-1)} +\cdots+ \ell x_{k}$. We denote the number $S_{\mathfrak{z},m}^{(\ell, \epsilon)}(k;r;v)$, which is defined to be the least positive integer $t$ such that for any $m$-coloring $\chi: [1, t] \to \{0, 1,\ldots,m-1\}$, there exists a solution $(\hat{x}_1, \hat{x}_2, \ldots, \hat{x}_k)$ to the equation $\mathcal{E}_{v}^{(\ell,\epsilon)}$ that satisfies the $r$-zero-sum condition, namely, $\displaystyle\sum_{i=1}^k\chi(\hat{x}_i) \equiv 0\pmod{r}$. In this article, we completely determine the constant $S_{\mathfrak{z}, 2}^{(k,1)}(k;r;0)$, $S_{\mathfrak{z}, m}^{(k-1,1)}(k;r;0)$, $S_{\mathfrak{z}, 2}^{(1,1)}(k;2;1)$ and $S_{\mathfrak{z}, r}^{(1,0)}(k;r;v)$. Also, we prove upper bound for the constants $S_{\mathfrak{z},2}^{(2,1)}(k;2;0)$ and $S_{\mathfrak{z},2}^{(1,1)}(k;2;v)$.