  We give an explicit description of hyperbolic Reinhardt domains D in C^2 such
that: (i) D has C^k-smooth boundary for some k greater than or equal to 1, (ii)
D intersects at least one of the coordinate complex lines $\{z_1=0\}$,
$\{z_2=0\}$, and (iii) D has noncompact automorphism group. We also give an
example that explains why such a setting is natural for the case of hyperbolic
domains and an example that indicates that the situation in C^n for n greater
than or equal to 3 is essentially more complicated than that in C^2.
