  Given any strictly convex norm $\|\cdot\|$ on $\mathbb{R}^2$ that is $C^1$ in
$\mathbb{R}^2\setminus\{0\}$, we study the generalized Aviles-Giga functional
\[I_{\epsilon}(m):=\int_{\Omega} \left(\epsilon \left|\nabla m\right|^2 +
\frac{1}{\epsilon}\left(1-\|m\|^2\right)^2\right) \, dx,\] for
$\Omega\subset\mathbb R^2$ and $m\colon\Omega\to\mathbb R^2$ satisfying
$\nabla\cdot m=0$. Using, as in the euclidean case $\|\cdot\|=|\cdot|$, the
concept of entropies for the limit equation $\|m\|=1$, $\nabla\cdot m=0$, we
obtain the following. First, we prove compactness in $L^p$ of sequences of
bounded energy. Second, we prove rigidity of zero-energy states (limits of
sequences of vanishing energy), generalizing and simplifying a result by
Bochard and Pegon. Third, we obtain optimal regularity estimates for limits of
sequences of bounded energy, in terms of their entropy productions. Fourth, in
the case of a limit map in $BV$, we show that lower bound provided by entropy
productions and upper bound provided by one-dimensional transition profiles are
of the same order. The first two points are analogous to what is known in the
euclidean case $\|\cdot\|=|\cdot|$, and the last two points are sensitive to
the anisotropy of the norm $\|\cdot\|$.
