  Let $\Delta$ be a rank 2 hyperbolic root system. Then $\Delta$ has
generalized Cartan matrix $H(a,b)= \left(\begin{smallmatrix} ~2 & -b\\ -a & ~2
\end{smallmatrix}\right)$ indexed by $a,b\in\mathbb{Z}$ with $ab\geq 5$. If
$a\neq b$, then $\Delta$ is non-symmetric and is generated by one long simple
root and one short simple root; whereas if $a= b$, $\Delta$ is symmetric and is
generated by two long simple roots. We prove that if $a\neq b$, then $\Delta$
contains an infinite family of symmetric rank 2 hyperbolic root subsystems
$H(k,k)$ for certain $k\geq 3$, generated by either two short or two long
simple roots. We also prove that $\Delta$ contains non-symmetric rank 2
hyperbolic root subsystems $H(a',b')$, for certain $a',b'\in\mathbb{Z}$ with
$a'b'\geq 5$. One of our tools is a characterization of the types of root
subsystems that are generated by a subset of roots. We classify these types of
subsystems in rank 2 hyperbolic root systems.
