II. (50 points) A newly built city has purchased a large number of public buses to solve the city's transportation problems. It plans to establish bus stops at 2009 different locations and connect them by opening several bus routes. The planners' wishes are:
(1) To open as many routes as possible;
(2) Every two routes should have at least one common bus stop; $\square$
(3) Each bus stop should have at most two different routes passing through it.

Question: According to these wishes, what is the maximum number of bus routes that can be opened? How many stops should each route have at a minimum?
Let $A$ denote the set of bus stops, then $|A|=2009$. Suppose a total of $m$ routes are opened, and the set of stops on each route is denoted as $A_{1}, A_{2}, \cdots, A_{m}$, with $A_{i} \subset A$.

From conditions (2) and (3), we have:
( i ) $A_{i} \cap A_{j} \neq \phi, 1 \leqslant i<j \leqslant m$;
( ii ) $A_{i} \cap A_{j} \cap A_{k}=\phi, 1 \leqslant i<j<k \leqslant m$.
That is, every pair of subsets intersects, but the intersection of any three subsets is empty.

From (i), we know that each subset shares at least one common element with the other $m-1$ subsets, and from (ii), we know that these common elements are all distinct. Therefore, the number of distinct elements in these $m$ subsets is at least $m(m-1)$. Hence, the number of elements in $\bigcup_{i=1}^{m} A_{i}$ is at least $\frac{m(m-1)}{2}$, but $\bigcup_{i=1}^{m} A_{i}=A$.
$$
\therefore \frac{m(m-1)}{2} \leqslant 2009 \text {. }
$$

The largest integer satisfying this condition is $m=63$.
Therefore, according to the planner's wishes, at most 63 bus routes can be opened, with each route passing through at least $m-1=62$ bus stops.