  Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and set
$\lambda: = \Lambda_1 - \Lambda_2$, where $\Lambda_1, \Lambda_2$ are the
fundamental weights for $\mathfrak{g}$; note that $\lambda$ is neither dominant
nor antidominant. Let $\mathbb{B}(\lambda)$ be the crystal of all
Lakshmibai-Seshadri paths of shape $\lambda$. We prove that (the crystal graph
of) $\mathbb{B}(\lambda)$ is connected. Furthermore, we give an explicit
description of Lakshmibai-Seshadri paths of shape $\lambda$.
