Do you think that there is a linear way to express this, using maybe a characteristic of this system that I cannot see. This is a set of equations involving exactly all sets of $(x_i, y_j, y_k)$ or $(x_i, x_j, y_k)$ with $i,j,k=1, ..., 3, i \ne j \ne k$):
$$1x_1 + 0x_2 + 0x_3 + 0y_1 + 1y_2 + 1y_3 = 0$$ $$\lor \,0x_1 + 1x_2 + 1x_3 + 1y_1 + 0y_2 + 0y_3 = 0$$ $$\lor \,0x_1 + 1x_2 + 0x_3 + 1y_1 + 0y_2 + 1y_3 = 0$$ $$\lor \,1x_1 + 0x_2 + 1x_3 + 0y_1 + 1y_2 + 0y_3 = 0$$ $$\lor \,0x_1 + 0x_2 + 1x_3 + 1y_1 + 1y_2 + 0y_3 = 0$$ $$\lor \,1x_1 + 1x_2 + 0x_3 + 0y_1 + 0y_2 + 1y_3 = 0$$
Knowing that :
$$x_1 \geq 0, x_2 \geq 0, x_3 \geq 0, y_1 \geq 0, y_2 \geq 0, y_3 \geq 0$$
By linear way I am talking about a set of linear equations or inequations that is equivalent, and could be put in a linear program (not a mixed integer program). The logical $or$ operator being not allowed in a linear program.