Geodesic
Holomorphic Equivalence of Tube Domains in~$\mathbb C^2$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and applications, Tome 253 (2006), pp. 101-110

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It is proved that, except in a few explicitly described cases, two connected tube domains in $\mathbb C^2$ are biholomorphically equivalent if and only if they are affinely equivalent. 
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     title = {Holomorphic {Equivalence} of {Tube} {Domains} in~$\mathbb C^2$},
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N. G. Kruzhilin; P. A. Soldatkin. Holomorphic Equivalence of Tube Domains in~$\mathbb C^2$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and applications, Tome 253 (2006), pp. 101-110. http://geodesic.mathdoc.fr/item/TM_2006_253_a8/