Zero-sum Generalized Schur Numbers

Let $r$ and $k$ be positive integers with $r\mid k$. Denote by $S_{\mathfrak{z}}(k;r)$ the minimum integer $n$ such that every coloring $\chi :[1,n]\to \{0,1,\dots ,r-1\}$ admits a solution to ${\sum }_{i=1}^{k-1}{x}_i=x_k$ with $\sum_{i=1}^{k} \chi(x_i) \equiv 0 ,($\mathrm{m}\mathrm{o}\mathrm{d}$,r)$. We give some formulas and lower bounds for various instances.