2. As shown in Figure 1, in the dihedral angle $\alpha-l-\beta$ of $120^{\circ}$, $\odot O_{1}$ and $\odot O_{2}$ are respectively
in the half-planes $\alpha, \beta$,
and are tangent to the edge $l$ at the same
point $P$. Then the sphere with
$\odot O_{1}$ and $\odot O_{2}$ as cross-sections ( ).
(A) has only 1
(B) has only 2
(C) has infinitely many
(D) does not exist
2. A

It is known that $\angle O_{1} P O_{2}$ is the plane angle of the dihedral angle $\alpha-l-\beta$.

In the plane $O_{1} \mathrm{PO}_{2}$, draw perpendiculars from points $O_{1}$ and $O_{2}$ to $P O_{1}$ and $\mathrm{PO}_{2}$, respectively, and let the intersection of these two perpendiculars be $O$. Since $\angle O_{1} P O_{2}=120^{\circ}$, such an intersection point $O$ is unique. Therefore, there is one and only one sphere with $\odot \mathrm{O}_{1}$ and $\odot \mathrm{O}_{2}$ as cross-sections, whose center is $O$ and radius is $O P$.