  In this manuscript we consider non-degenerate surfaces $\Sigma^2$ immersed in
a 3-dimensional homogeneous space $\mathbb{L}^3(\kappa,\tau)$ endowed with two
different metrics, the one induced by the Riemannian metric of
$\mathbb{E}^3(\kappa,\tau)$ and the non-degenerate metric inherited by the
Lorentzian one of $\mathbb{L}^3(\kappa,\tau)$. Therefore, we have two different
geometries on $\Sigma^2$ and we can compare them. In particular, we can
consider the Gaussian curvature functions which respect to both metrics and
study the geometry of the surfaces satisfying that both Gaussian curvature
functions are opposite. We will call these surfaces anisocurved surfaces. In
order to obtain our main results we also need to impose some extra assumptions
regarding the extrinsic curvatures with respect to both metrics.
