Geodesic
Topology of Real Algebraic Curves
Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 23-27

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The problem on the existence of an additional first integral of the equations of geodesics on noncompact algebraic surfaces is considered. This problem was discussed as early as by Riemann and Darboux. We indicate coarse obstructions to integrability, which are related to the topology of the real algebraic curve obtained as the line of intersection of such a surface with a sphere of large radius. Some yet unsolved problems are discussed.
Keywords: geodesic flow, analytic first integral, geodesic convexity, $M$-curve. 
@article{FAA_2008_42_2_a3,
     author = {V. V. Kozlov},
     title = {Topology of {Real} {Algebraic} {Curves}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {23--27},
     publisher = {mathdoc},
     volume = {42},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a3/}
}
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V. V. Kozlov. Topology of Real Algebraic Curves. Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 23-27. http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a3/