I was wondering whether the exists a well-known solution for solving non-linear systems in the following form for each of the $n$ variables $x_i$:
$$\begin{cases}\frac{1}{\mathbf{1}_{C_{i}\neq 0}+|S_{i}|}\left (C_{i}+\sum_{k\in S_{i}}\prod_{j=1}^n B_{kj}x_j\right)=x_{i} & S_i\neq \emptyset \\ x_i = C_i & \textrm{oth.}\end{cases}$$
where $S_{i}\in\wp(\{1,\dots,n\})$ , $0\leq C_{\alpha_i}\leq 1$, $B_{ij}=0\vee B_{ij}=1$, and $\mathbf{1}_p$ is the indicator function returning $1$ when $p$ holds and 0 otherwise.
An example of such a system would be the following one:
$$\begin{cases} \frac{1}{2}(\frac{1}{2}+x)=y\\ \frac{1}{2}(\frac{3}{10}+xy)=z\\ xyz = t\\ \frac{1}{2}(1+ty)=x \end{cases}$$
Given that my field of research is not mathematics, I would not even know where to start looking for this as, if such kinds of systems are known, I don't know a name for these. If, on the other hand, such systems are known, I would like to know whether some known resolution methods might also be computed by a computer.