Spectral form factor in the Hadamard-Gutzwiller model: Orbit pairs contributing in the third order

In this paper we consider periodic orbit pairs contributing in the third order of the spectral form factor in the Hadamard-Gutzwiller model. We prove that periodic orbits involving two 2-encounters in certain structures have partner orbits, which together with original ones form orbit pairs and contribute in the third order of the spectral form factor. The action differences are estimated at $\ln (1+u_1{s}_1)(1+u_2{s}_2)$ with explicit error bounds, where $(u_1,s_1)$ and $(u_2,s_2)$ are the coordinates of the piercing points. A symbolic dynamics for orbit pairs via conjugacy classes is also established.