Geodesic
Transcendental first integrals of dynamical systems on the tangent bundle to the sphere
Contemporary Mathematics and Its Applications, Tome 100 (2016), pp. 58-75.

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In this paper, we examine the existence of transcendental first integrals for some classes of systems with symmetries. We obtain sufficient conditions of existence of first integrals of second-order nonautonomous homogeneous systems that are transcendental functions (in the sense of the theory of elementary functions and in the sense of complex analysis) expressed as finite combinations of elementary functions. 
@article{CMA_2016_100_a5,
     author = {M. V. Shamolin},
     title = {Transcendental first integrals of dynamical systems on the tangent bundle to the sphere},
     journal = {Contemporary Mathematics and Its Applications},
     pages = {58--75},
     publisher = {mathdoc},
     volume = {100},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMA_2016_100_a5/}
}
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M. V. Shamolin. Transcendental first integrals of dynamical systems on the tangent bundle to the sphere. Contemporary Mathematics and Its Applications, Tome 100 (2016), pp. 58-75. http://geodesic.mathdoc.fr/item/CMA_2016_100_a5/