(The polar equation of the hyperbola is
$$\rho=\frac{2}{1-\sqrt{3} \cos \theta} \text {. }$$

Let $A B$ be a chord passing through the right focus and intersecting only the right branch.
Then $|A B|=\rho_{1}+\rho_{2}$
$$\begin{array}{l} =\frac{2}{1-\sqrt{3} \cos \theta}+\frac{2}{1-\sqrt{3} \cos (\theta+\pi)} \\ =\frac{4}{1-3 \cos ^{2} \theta} \geqslant 4 \end{array}$$
(when $\theta=\frac{\pi}{2}$, the equality holds), that is, among the chords passing through the right focus of the hyperbola and intersecting two points on the right branch, the minimum length is 4 if and only if the chord is perpendicular to the $x$-axis. Since there are exactly 3 lines that satisfy the condition, we have
(1) There is only one line that intersects both the left and right branches of the hyperbola, which must be the real axis of the hyperbola by symmetry, and there are only two lines that intersect the right branch, which can be verified not to meet the condition;
(2) There are only two lines that intersect both the left and right branches of the hyperbola, and only one line that intersects the right branch, which must be perpendicular to the $x$-axis by symmetry. In this case, $|A B|=\lambda=4$. When $\lambda=4$, it can be proven that there are two lines that intersect both the left and right branches of the hyperbola and have a length of 4, so $\lambda=4$. )