We investigate domains $\Omega = \{\mathrm {Re}\, w + Q(z,\overline {z}) < 0\}$ in $\mathbb {C}^2$ where $Q$ is a subharmonic and non-harmonic polynomial. The holomorphic equivalence of domains is characterized in terms of the defining polynomials. In the case that $\mathrm {dim}_R\,\mathrm {Aut}(\Omega ) \geq 2$, we give canonical defining polynomial and we list all possible Aut($\Omega $).