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Parity-time (PT) symmetry was first studied in quantum mechanical systems with a non-Hermitian Hamiltonian whose observables are real-valued. Most existing designs of PT symmetric systems in electronics, optics, and acoustics rely on an exact balance of loss and gain in the media to achieve PT symmetry. However, the dispersive behavior of most loss and gain materials restricts the frequency range where the system is PT symmetric. This makes it challenging to access the exceptional points of the system to observe the PT symmetric transition dynamics. Here, we propose a new path to realize PT symmetric systems based on gyroscopic effects instead of using loss and gain units. We demonstrate that PT symmetry and the occurrence of exceptional points are preserved for inversive, counter-rotating gyroscopic systems even with dispersive sub-units. In a gyroscopic system with two circular rings rotating in opposite directions at the same speed, the spontaneous symmetry breaking across the exceptional points results in a phase transition from a moving maximum deformation location to a motionless maximum point. The motionless maximum point occurs despite the externally imposed rotation of the two rings. The results set the foundation to study nonlinear dispersive physics in PT symmetric systems, including solitary waves and inelastic wave scattering.