Geodesic
Algebro-topological approach to reality problems. Real action variables in the theory of finite-gap solutions of the Sine-Gordon equations
Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part VI, Tome 133 (1984), pp. 177-196

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The paper developes an algebro-topological approach to the problem of effective selection of real finite gap solutions of the sine-Gordon equation, based on the so-called $\gamma$-representation associated with a Riemann surface where action variables can be written in a closed form. The approach is a general one and applies to many other systems for which the reality problem has not yet been solved. 
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     author = {S. P. Novikov},
     title = {Algebro-topological approach to reality problems. {Real} action variables in the theory of finite-gap solutions of the {Sine-Gordon} equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {177--196},
     publisher = {mathdoc},
     volume = {133},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_133_a12/}
}
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S. P. Novikov. Algebro-topological approach to reality problems. Real action variables in the theory of finite-gap solutions of the Sine-Gordon equations. Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part VI, Tome 133 (1984), pp. 177-196. http://geodesic.mathdoc.fr/item/ZNSL_1984_133_a12/