  Let $M$ be a differentiable manifold and $TM$ be its tangent bundle. A
$C^0$-Finsler structure on $M$ is a continuous function $F: TM \rightarrow
\mathbb R$ such that its restriction to each tangent space is a norm. In this
work we present a large family of projectively equivalent $C^0$-Finsler
manifolds $(\hat M \cong \mathbb R^2,\hat F)$. Their structures $\hat F$ don't
have partial derivatives and they aren't invariant by any transformation group
of $\hat M$. For every $p,q \in (\hat M,\hat F)$, we determine the unique
minimizing path connecting $p$ and $q$. They are line segments parallel to the
vectors $(\sqrt{3}/2,1/2)$, $(0,1)$ or $(-\sqrt{3}/2,1/2)$, or else a
concatenation of two of these line segments. Moreover $(\hat M,\hat F)$ aren't
Busemann $G$-spaces and they don't admit any bounded open $\hat F$-strongly
convex subsets. Other geodesic properties of $(\hat M,\hat F)$ are also
studied.
