Geodesic
On~complex structures on two-dimensional tori admitting metrics with nontrivial quadratic integral
Matematičeskie zametki, Tome 58 (1995) no. 5, pp. 643-652

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In this paper the set of complex structures on a torus admitting Riemannian metrics (consistent with these complex structures) with nontrivial quadratic integrals is described. 
@article{MZM_1995_58_5_a0,
     author = {I. K. Babenko and N. N. Nekhoroshev},
     title = {On~complex structures on two-dimensional tori admitting metrics with nontrivial quadratic integral},
     journal = {Matemati\v{c}eskie zametki},
     pages = {643--652},
     publisher = {mathdoc},
     volume = {58},
     number = {5},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1995_58_5_a0/}
}
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I. K. Babenko; N. N. Nekhoroshev. On~complex structures on two-dimensional tori admitting metrics with nontrivial quadratic integral. Matematičeskie zametki, Tome 58 (1995) no. 5, pp. 643-652. http://geodesic.mathdoc.fr/item/MZM_1995_58_5_a0/