I wonder what could be a Kähler surface, which is a total space of principal elliptic bundle over a curve. I believe that there is a classification and that it must be pretty simple, but I cannot find it in the literature.

Fo example, though there exist a full and complicated classification of elliptic surfaces, I didn't manage to find the list of all elliptic surfaces with no singular fibers.

Is it true, for example, that any of such surface is covered by the product of the base and the fiber? In [Höfer, Thomas, Remarks on torus principal bundles, J. Math. Kyoto Univ. 33 (1993), no. 1, 227–259.] it is proved, that such a fibration must be flat (in the following sence: it is covered by a flat principal $\mathbb{C}^*$-bundle), so it is enough to show that it has finite monodromy...