Geodesic
An indeterminate equation
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 4, Tome 67 (1977), pp. 163-166

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It is proved that on the curve $$ x_0^2+x_1^2=t(x^2_2-x_3^2),\quad t(x_0^2-x_1^2)=x_2^2+x_3^2 $$ there are no $k(t)$ – rational points; here $k$ is an algebraically closed field. 
@article{ZNSL_1977_67_a7,
     author = {V. A. Dem'yanenko},
     title = {An indeterminate equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {163--166},
     publisher = {mathdoc},
     volume = {67},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_67_a7/}
}
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V. A. Dem'yanenko. An indeterminate equation. Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 4, Tome 67 (1977), pp. 163-166. http://geodesic.mathdoc.fr/item/ZNSL_1977_67_a7/