Geodesic
Affine and Holomorphic Equivalence of Tube Domains in $\mathbb C^2$
Matematičeskie zametki, Tome 75 (2004) no. 5, pp. 670-682

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It is shown that, except for several explicitly described cases, two hyperbolic tube domains in $\mathbb C^2$ are biholomorphically equivalent if and only if they are affinely equivalent. 
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     author = {N. G. Kruzhilin and P. A. Soldatkin},
     title = {Affine and {Holomorphic} {Equivalence} of {Tube} {Domains} in $\mathbb C^2$},
     journal = {Matemati\v{c}eskie zametki},
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N. G. Kruzhilin; P. A. Soldatkin. Affine and Holomorphic Equivalence of Tube Domains in $\mathbb C^2$. Matematičeskie zametki, Tome 75 (2004) no. 5, pp. 670-682. http://geodesic.mathdoc.fr/item/MZM_2004_75_5_a3/