Geodesic
Holomorphic equivalence of Reinhardt domains in~$\mathbb C^2$
Izvestiya. Mathematics , Tome 66 (2002) no. 6, pp. 1271-1304.

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We consider the following question: are any two holomorphically equivalent Reinhardt domains necessarily algebraically equivalent? The answer is known to be positive in the one-dimensional case. We get a positive answer in dimension two. Apart from connectedness, no restrictions are imposed on the domains. 
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P. A. Soldatkin. Holomorphic equivalence of Reinhardt domains in~$\mathbb C^2$. Izvestiya. Mathematics , Tome 66 (2002) no. 6, pp. 1271-1304. http://geodesic.mathdoc.fr/item/IM2_2002_66_6_a7/

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