19. Answer: 12 swans

Analysis:

Method one: (Arithmetic method) From "a pair of swans more than a lake", we know the number of swans is a multiple of 2; from "three swans less than a lake", we know the number of swans is a multiple of 3. Since $(2,3)=1$, we know the number of swans is a multiple of 6: $6, 12, 18, 24, \cdots$ Verification yields: a total of 12 swans landed.

Method two: (Algebraic method) Let the number of swans be $\mathrm{x}$, and they land in $\mathrm{Y}$ lakes. Then $2 \mathrm{y}+2=3 y-3$,

Solving for $y$ gives $y=5$, then $x=2 y+2=2 \times 5+2=12$, i.e., a total of 12 swans landed.
Ego City's West Lake falls with swans, one lake one pair two swans more, one lake three three less, how many in total have fallen?
(Explanation: Huizhou is nicknamed "Ego City", and the West Lake in the city is a famous scenic area, consisting of 5 lakes: Feng, E, Ping, Ling, and Nan. The question means: one lake has one pair of swans with two swans more, one lake has three swans with three swans less, how many swans have fallen in total?)