Geodesic
Ergodic perturbations of degenerate integrable Hamiltonian systems
Izvestiya. Mathematics , Tome 7 (1973) no. 3, pp. 535-571

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Hamiltonian systems arbitrarily close in the $C^r$ topology $(r=1,2,\dots)$ to a given integrable degenerate Hamiltonian system of class $C^\infty$ which generate an ergodic flow on each manifold of constant energy are constructed. Applications: Small perturbations of a system generated by independent oscillators and Finsler metrics close to standard Riemannian metrics on symmetric spaces of rank 1. 
@article{IM2_1973_7_3_a4,
     author = {A. B. Katok},
     title = {Ergodic perturbations of degenerate integrable {Hamiltonian} systems},
     journal = {Izvestiya. Mathematics },
     pages = {535--571},
     publisher = {mathdoc},
     volume = {7},
     number = {3},
     year = {1973},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1973_7_3_a4/}
}
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A. B. Katok. Ergodic perturbations of degenerate integrable Hamiltonian systems. Izvestiya. Mathematics , Tome 7 (1973) no. 3, pp. 535-571. http://geodesic.mathdoc.fr/item/IM2_1973_7_3_a4/