2. Through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$, a line $l$ intersects the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=$ $\lambda$, and there are exactly 3 such lines $l$, then $\lambda=$ $\qquad$.
2. 4 .

First, note the following conclusion: A chord passing through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$ and intersecting the right branch at two points attains its minimum length $\frac{2 b^{2}}{a}=4$ if and only if the chord is perpendicular to the $x$-axis. (In fact, the polar equation of this hyperbola is $\rho=\frac{2}{1-\sqrt{3} \cos \theta^{\circ}}$. Suppose $A B$ is a chord passing through the right focus $F$ and intersecting only the right branch, with $A\left(\rho_{1}, \theta\right), B\left(\rho_{1}, \pi+\theta\right)\left(\rho_{1}>0, \rho_{2}>0\right)$. Then $|A B|=\rho_{1}+\rho_{2}=\frac{2}{1-\sqrt{3} \cos \theta}+\frac{2}{1+\sqrt{3} \cos \theta}=\frac{4}{1-\sqrt{3} \cos ^{2} \theta} \geqslant 4$, with equality when $\theta=\frac{\pi}{2}$.)
Second, when there are three lines satisfying the given conditions, there are only two possibilities:
(1) Only one line intersects both the left and right branches of the hyperbola, while two lines intersect only the right branch. In this case, the line intersecting both branches must be the $x$-axis, and the distance between its two intersection points is $2 a=2$. However, the length of the two chords intersecting only the right branch, $\lambda>4$, does not satisfy the given conditions;
(2) Two lines intersect both the left and right branches of the hyperbola, while only one line intersects only the right branch. This chord must be perpendicular to the $x$-axis (otherwise, by symmetry, there would be two chords intersecting only the right branch). In this case, $|A B|=\lambda=4$, and the length of the chords intersecting both branches can also satisfy this condition, so $\lambda=4$.