  We construct an integrable hierarchy in the form of Hirota quadratic
equations (HQE) that governs the Gromov--Witten (GW) invariants of the Fano
orbifold projective curve $\mathbb{P}^1_{a_1,a_2,a_3}$. The vertex operators in
our construction are given in terms of the $K$-theory of
$\mathbb{P}^1_{a_1,a_2,a_3}$ via Iritani's $\Gamma$-class modification of the
Chern character map. We also identify our HQEs with an appropriate
Kac--Wakimoto hierarchy of ADE type. In particular, we obtain a generalization
of the famous Toda conjecture about the GW invariants of $\mathbb{P}^1$ .
