Identify Smith–Van den Bergh rulings for A = S/(x1^2+x2^2+x3x4+x4x3)

Determine, for A = S/(x_1^2 + x_2^2 + x_3 x_4 + x_4 x_3), the ruling(s) on Proj A in the sense of Smith–Van den Bergh corresponding to each indecomposable maximal Cohen–Macaulay module M in the set ℳ = { M ∈ MCM(A) : M is indecomposable, M_0 ≅ K^2, and M = M_0 A }.

Background

The algebra S has a central quadratic element z = x_1^2 + x_2^2 + x_3 x_4 + x_4 x_3, and the quotient A = S/z is shown to be a graded isolated singularity. Smith and Van den Bergh define rulings on noncommutative quadric surfaces Proj A via indecomposable maximal Cohen–Macaulay modules.

This question requests the explicit identification of the ruling(s) associated with each module in the set ℳ, bridging the homological definition of rulings with the geometric configurations studied for S.