Eternal Engine is the 13th stage in the Hero story on Sonic Adventure 2. Tails is used to play this stage.
There are two vastly different ways to play this level: a "natural" path that somewhat resembles a level, and an "exploit" path that skips practically everything and turns the level into one massive hover session. Due to the difficulties involved in the exploit, both paths will be explained.
A version difference at the end here favors the Dreamcast version by about 15 seconds.
Lock on to one of the beetle enemies guarding the first door as you're falling, then shoot the second one as you land. Run through the door and down the hallway, locking on to the two enemies at the second locked door. You don't need to shoot out the box to make a staircase; just jump at the spring from the side and you'll be launched up to the pulley. The time you want to beat at the spring is 11.1 seconds; failure to do so means you'll have to wait for the circling beetle to pass or hover around it, costing more time. Go to the second pulley, and jump and hover forward as soon as the screen jitters to indicate you've reached the top.
You can shoot out the Hornet enemies in this hall; most enemies are optional to hit if you prefer to dodge instead. When the door starts to open, release forward for a bit before picking it back up, to delay your arrival at the door until there's enough space for you to squeeze through (as opposed to running into the door when there isn't enough space, losing all momentum). You'll repeat this method for the remaining doors in the stage. Arrive at the checkpoint; high 00:21 is possible if everything goes right, but you'll normally get 22 or 23.
Another optional shooting hall leads you to a room with doors on three sides. Shoot out the Hornet in front of you and start firing volkan cannon shots straight through the door, before it opens and you end up locking onto dynamite. If you hear the sound of a chao box being shot out, that's good. It means you can either jump over a single box in the next room and tag the switch, or better yet avoid jumping and just run over it. Once you hit the switch, jump and turn around quickly in midair, then head back out of the room and through the door you unlocked.
Here's your first venture along space catwalks. Hug the left side and lock onto the Hornet. Residual damage will take out the ground below it, but that's okay. Just keep running at full speed, hover over the gap, and the beetle will just miss you. Through another hallway, you'll come to a balcony. Jump off the edge here, being sure to land on the ground below and not in empty space for a fall out. Stand on the green button, and as soon as you see smoke, aim for the middle, farthest away of the three moving platforms. If it's really far away in its cycle, you might have to jump on the rails of your catwalk to get enough height to reach it. Jump again over the remaining metal cages.
As soon as the next door opens, lock on to the tank up ahead. You need to shoot it 4 times, and there's just enough time to do that. When you see the shadow of the weight, jump before it and stay in a hover as you go past. The weight doesn't care that you're underneath it in air; it only drops if you're on the ground near it. Hover until you reach the now-exposed switch and hit it, and the weight never gets a chance to drop! Spring up and hover at the peak of the bounce to land on the weight and get to the pulley. Ride it up to the second pulley and hit the checkpoint.
There are some moving platforms up ahead that move on a 9-second cycle. If you got to this checkpoint at 01:11, the platforms will be in the favorable part of their cycle if you keep going at full speed. If you got there faster, you'll have to wait on them, unless you die and restart from that checkpoint. Shoot out the Chaos enemy as soon as it starts to fire lasers, so you can unlock the door. In the next hallway, there's another Chaos. You don't need to kill it, but go ahead and lock onto it once. This will delay its tentacles until you're safely out of reach of getting hit.
Enter the platform room. Jump up using boxes as a staircase, and from there to the first platform on its way up. Jump, fire a volkan cannon shot into the balloon, and hover, and the balloon hit will give you and upward boost that allows you to hover to the other side. Hit the checkpoint (the time should be around 01:25) and head out to the multi-layered catwalks.
The path is easy to follow here, but try to cut corners where possible to save time, and try not to go overboard on the shooting or else you might knock out the ground from underneath yourself. Re-enter the colony at the top and hit the checkpoint, with 01:50 or better. Another hallway leads you out to the next catwalk. Just jump off to one side and land on the platform below. There's a wall of boxes, and you need to shoot out a stack of 2 boxes in order to pass through (preferably 2 adjacent stacks to avoid getting stuck).
Jump through the opening you created, and continue down the hallway into the next room. There's a Chaos here, but again just shoot it so it hides in its shell until you're safely past. Run to the pulley, jump back to the next pulley, and forward to the plaform. Turn the corner and hit the last checkpoint (preferably at 02:22 or lower).
Here's where the version difference comes in. There's an updraft in the open space ahead, but it only extends so far left and right. The colony has walls that extend past the updraft range in either direction. However, on Dreamcast the walls outside that range are actually fake, and can be passed through harmlessly. The strategy there is to jump on the rails of the platform ahead, immediately jump and hover to the outside of the updraft range and continue forward, passing through the fake wall. Then release hover and fall, maneuvering around to the back of the digital wall without passing back through it (confining yourself to within boundaries again). If you went too far back, you'll either die by fall out or get an endless fall that makes you restart the whole level, and if you didn't go back far enough you'll land on the roof of the power generator room. If you're in just the right position, though, you'll land directly on the goal ring without destroying the power generator, in a time of under 2:40.
GameCube players have to take the traditional route going down the long shaft and destroying the spots on the generator, but can still aim for 3:00.
Best video using this path: 2:29.72 by Psyknux
Stay on course up to the first checkpoint. Walk down the hallway, and when you get to the fork that lets you move forward to hit a switch that unlocks the path to your right, stop. Now move left, up against the wall.
There's a hatch here that has no dynamite on it. Notice the red lines at the edge; you want to stand near the red line closest to the path you just came from. Push into the wall attempting to stay on the line, and...sometimes you inexplicably fall through the floor. You don't die, you just...fall through the floor. Now you can hover. The path to your right obviously isn't blocked anymore; it's just open space, but if you try to approach it directly you'll hit a kill plane and die. Better to go around it first, or at least drop down below it beforehand. As you keep going, you'll find that you've sunk below all the surfaces you can land on, so things may look bleak. Don't worry, just keep on going.
By the way, if you ever attempt this path, it helps to learn the general shape of Eternal Engine. You won't be able to see all of it what with how low you've sunk, but it will vastly improve your sense of direction.
Eventually, after several seconds where you have no guiding landmarks whatsoever--only the background--if you headed in the right direction, you may come across a tower. Unlike everything else, this extends low enough that you can reach it. And wouldn't you know...it's the tower at the very end of the level! You can either reenter it and try to take out the generator, or you can hover around to one side and attempt to hit the goal ring from the top (note, though, that given the different angle of approach it's a lot harder to do this than it is with the Dreamcast shortcut on the natural path). Pick your path and finish the level faster than ever.
Record video: 1:31.72 by SadisticMystic
Play out the "natural" mission 1 path, staying in the center of hallways to collect rings. There are 21 before the first checkpoint, for a bonus of 5 (plus 5 in the same room for 31); 47 before the second checkpoint for a bonus of 10; and 65 by the third for a bonus of 20.
Now as you navigate the catwalks, on the second level up, you'll notice a place you can drop down and hit a switch. Do that, then hover back to the start of the catwalks and go back in the level, toward checkpoint 3. You'll notice that a door got opened. In that room, behind black boxes, you'll find a 1-up on the left and 20 rings on the right. Make a note of the position of the 20 rings.
On a speedrun, you're not going to go through all that. Instead, you'll fire volkan cannon shots through the closed door and hit the 20 right away. It's tough, being a blind shot, but it's necessary to get in the low 1:30s or better.
Best video: 1:26.50 by Psyknux
Record video: 1:00.37 by Psyknux
Speedrun-wise, this plays almost identically to mission 1. The only changes in strategy are that the box covering the first spring has to be shot out to give you enough height on the bounce, and that the first switch (if you elect to use the natural path) has moved from behind the center boxes to underneath the left box. Other than that, the changes just require more precision to pass or add more dynamite to discourage random shooting.
Best video: 2:32.22 by Psyknux