Symmetric Limit Cycles in 3D Piecewise Linear Systems with Visible-visible Two-Fold Singularity

We analyze a three-dimensional discontinuous piecewise linear system $Z=(X,Y)$ whose switching manifold $\Sigma$ contains visible-visible two-fold intersection lines. Assuming that the matrices $DX$ and $DY$ each have one nonzero real eigenvalue and one pair of complex conjugate eigenvalues, we reduce the system to a canonical form. Under a resonant condition, we use Darboux integrability theory to obtain a first integral common to $X$ and $Y$. Its restriction to $\Sigma$ defines a hyperbola $\Gamma$, which parametrizes the crossing points of symmetric periodic orbits. On this curve we construct the half-return maps, derive analytic expansions for the corresponding return times near infinity, and introduce a time-matching function given by their difference. By means of the Weierstrass Preparation Theorem, we prove the existence of a large-amplitude symmetric limit cycle for a suitable subfamily of systems. We then study stability through a saltation-corrected monodromy matrix and reduce the problem to Schur--Cohn inequalities for the two transverse Floquet multipliers.