If does it exists, how is regulated the invasion of lane in swimming?
It is allowed to "invade" the next lane during the stroke under water?
I'll go with the USA Swimming 2015 Rulebook (available on line at usaswimming.org under the Officials resources). These rules are in line with FINA international rules, so if not in the USA, your national Rulebook should still contain similar language.
USA Swimming rule 102.22.4 states:
A swimmer must start and finish the race in the same lane.
So, should they end up in a different land during the race, they must get back in to their lane for it to be a legal swim.
Furthermore, USA Swimming 102.22.6 states:
Obstructing or otherwise interfering with another swimmer shall disqualify the offender, subject to the discretion of the Referee.
So, if the lane you find yourself in has another swimmer in it, you can't get in their way. The 'discretion of the Referee' part is primarily there to give the Referee discretion as to who interfered with whom in less clear-cut cases, e.g. two backstrokers in adjacent lands tangling up with each other under the lane line.)
Now, suppose you ended up in the wrong lane, following the other swimmer - to this point you have not interfered with them (until they turn and come back your way). Or, suppose it was on a 50 long course race, where they won't come back your way. To avoid being disqualified under 102.22.6, you need to get back in to your lane, under the lane line, before you run into them. The trick is how to do that legally.
In breaststroke (USA Swimming rules 101.2) it would be fairly straightforward as part of the normal, legal stroke - your head goes under the lane line, you kick, and your head breaks the surface back in your lane.
In butterfly (USA Swimming rules 101.3) it is harder to see how - 101.3.2 includes the words
The swimmer must remain on the surface until the next turn or finish.
As an official, if a butterflier ducked under the lane line this would disqualify them.
In backstroke (USA Swimming rules 101.4), there is also wording (101.4.2) of
Some part of the swimmer must break the surface of the water throughout the race, except it is permissible for the swimmer to be completely submerged during the turn and for a distance of not more than 15 meters (16.4 yards) after the start and after each turn.
Freestyle (USA Swimming 101.5.2) has similar language to the backstroke.
Except perhaps for breaststroke, once you are swimming in the wrong lane, your only chance to get back in to your appropriate lane is on a turn, where the 15 meter rule applies.
Plausible scenario: On the start, within the 15 meter rule, you dolphin kick out in front of the adjacent swimmer, coming up well ahead. You realize your mistake. At the turn, you get back in to your lane, avoiding any contact with the other swimmer. However, at best, you are leaving it up to the Referee to determine whether or not your wake is interfering with the other swimmer (102.22.6). As an official I would call it in as a possible infraction, which would then lead to discussion with the Referee. Except under highly unusual circumstances, it would likely be a disqualification.
One of those unusual circumstances would be if there were no swimmer in the adjacent lane - then you have no possibility of interference, you would switch back to the correct lane on the turn, and life would go on. But then, would it really be an 'invasion'?