Normally, in solving problems with linear systems, one is given a linear system (perhaps with constraints) and wishes to find a solution, but in my case, I am given a solution and want to determine the existence of a linear system with particular structure, for which the given values are a solution.
That is, for $k_i, l_i \in \mathbb{Z}^+$, $\alpha, \beta \in (0, \pi/2)$ and $y_i \in (0, \pi/2]$, I want to determine existence of a system of form $$ y_0 + y_1 = \pi + k_1\alpha - l_1\beta \\ y_1 + y_2 = \pi + k_2\alpha - l_2\beta \\ \vdots \\ y_{n-2} + y_{n-1} = \pi + k_{n-1}\alpha - l_{n-1}\beta \\ y_{n-1} + y_0 = \pi + k_n\alpha - l_n\beta \\ $$ where $\alpha, \beta, y_0$ are known, and each equality satisfies the constraints given by $y_{i-1} > k_i \alpha$ and $\pi > l_i \beta + y_i$ for $i=1, 2, ..., n$ (in the last equality treat $y_0$ as $y_n$). For example, if $y_0 = \pi/2$, then for any $\alpha, \beta$ does there exist a system of the above form which also satisfies the inequality constraints?
