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In the present work the classical problem of harmonic oscillator in thehyperbolic space $H_2^2$: $z_0^2+z_1^2-z_2^2-z_3^2=R^2$ has been completelysolved in framework of Hamilton-Jacobi equation. We have shown that theharmonic oscillator on $H_2^2$, as in the other spaces with constant curvature,is exactly solvable and belongs to the class of maximally superintegrablesystem. We have proved that all the bounded classical trajectories are closedand periodic. The orbits of motion are ellipses or circles for bounded motionand ultraellipses or equidistant curve for infinite ones.

Harmonic Oscillator on the ${\rm SO}(2,2)$ Hyperboloid