Quantum integrable systems related with symmetric spaces of the groups U(1,2) and Sp(1,2) and Green functions on these spacesIntegrable quantum systems related with two-dimensional two-and one-sheeted hyperboloid are considered. The explicit expressions of the Green's functions of the free particles on the those homogenous spaces are given.

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Quantum integrable systems related with SL(2,R) group manifold in various coordinate systems defined by the various decompositions: KHK, HAH, HKH, NHN, NHK of the group SL(2,R) on one parameter subgroups K,H,A,N are considered. The explicit expressions for waves functions, spectra, and S-matrices are given.