Geodesic
Integrable motions of a pendulum in a two-dimensional plane
Contemporary Mathematics and Its Applications, Tome 100 (2016), pp. 36-57

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In this paper, we examine new cases of integrability of dynamical systems on the tangent bundle to a low-dimensional sphere, including flat dynamical systems that describe a rigid body in a nonconservative force field. The problems studied are described by dynamical systems with variable dissipation with zero mean. We detect cases of integrability of equations of motion in transcendental functions (in terms of classification of singularity) that are expressed through finite combinations of elementary functions. 
@article{CMA_2016_100_a4,
     author = {M. V. Shamolin},
     title = {Integrable motions of a pendulum in a two-dimensional plane},
     journal = {Contemporary Mathematics and Its Applications},
     pages = {36--57},
     publisher = {mathdoc},
     volume = {100},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMA_2016_100_a4/}
}
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M. V. Shamolin. Integrable motions of a pendulum in a two-dimensional plane. Contemporary Mathematics and Its Applications, Tome 100 (2016), pp. 36-57. http://geodesic.mathdoc.fr/item/CMA_2016_100_a4/