Consider the parametrization of a surface $\gamma(t,s)=(t+i\kappa_1(t,s),s+i\kappa_2(t,s))$ in $\mathbb{C}^2$, and let
\begin{equation} \mathcal{I}=\int dt ds \ \text{det}(\mathbb{J}\gamma)\frac{1}{f(\gamma(t,s))} \end{equation} be the integral of a complex function along the surface. Assume that the complex function $f(z), \ z=(z_1,z_2)$, $z_1,z_2\in\mathbb{C}$ has zeros on a bounded, connected surface which has no purely real points, denoted by $\mathcal{S}=\{z\in\mathbb{C}^2\ | \ f(z)=0, \ \mathcal{I}\text{m}(z)\neq 0\}$. We can say that the surface is included in the contour $\gamma$ if $\mathcal{S}$ lies in between the real plane and the surface parametrized by $\gamma$.
What is the condition, preferably expressed in terms of $f$ which, for a given surface $\gamma$, establishes that the surface $\mathcal{S}$ is or isn't included in the contour $\gamma$?