Systems of equations of the form $\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a$

Question

Is there any theory that deals (directly or not) with systems of equations of the form $$\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a,$$ where $a \in \mathbb{R}^3$ is known, $v_i, v_j \in \mathbb{R}^3$ are unknowns, and $\times$ denotes the cross product? $I$ and $J_i$ are sets of indices, with $J_i$ depending on $i$.