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We have studied optical metrics via null geodesics and optical-mechanicalformulation of classical mechanics, and described the geometry and optics ofmechanical systems with drag dependent quadratically on velocity. Then westudied null geodesics as a central force system, deduced the related Binet'sequation applied the analysis to other solutions of Einstein's equations inspherically symmetric spaces, paying special attention to the Tomimatsu-Satometric. Finally, we examined the dualities between different systems arisingfrom conformal transformations that preserve the Jacobi metric.

Dynamical systems of null geodesics and solutions of Tomimatsu-Sato 2