Consider the following equations
$$ \left\{ \begin{array}{} x=\frac{1}{n}\|\mathrm v-y1_n\|_1\\ y=\frac{1}{n}(\mathrm v^T 1_n-x1_n^T sign(\mathrm v-y1_n))\\ \end{array} \right. $$
where $x\in R^{+}$, $y\in R$, $\mathrm v \in R^n$, and $1_n$ denotes a vector of dimension $n$ with entries all equal to one. Also $sign(\mathrm w)$ returns a vector of the same size of its argument containing the sign of vector's elements.
I have tried an iterative approach $\left\{ \begin{array}{} x_{k+1}=\frac{1}{n}\|\mathrm v-y_k1_n\|_1\\ y_{k+1}=\frac{1}{n}(\mathrm v^T 1_n-x_k1_n^T sign(\mathrm v-y_k1_n))\\ \end{array} \right.$ to solve this system. However, depending on initial values $x_0$ and $y_0$, solutions may be converged to different values.
How can I find the number of possible solutions to the above system? and is it possible to have these solutions in a closed-form?