The Blackest Heart solution: thrust into one of the vortexes at a diagonal from your starting point, being at either point A (click on photo below) or B. You will emerge at either the vortex marked with a 1 or 2 depending on which one you chose (i. e. if you chose vortex A to start with, you will appear at vortex 1, if you chose vortex B, you will appear at vortex 2). Thrust down to bottom middle vortex C, then repeat at the opposite side, hitting either the lower vortexes of 1 or 2, which will send you to vortex A or B and complete the puzzle.
The Judas Paradox: there are several possible ways to solve this one, being two short and one long way.
The shortest ways involve thrusting in a straight line, hitting vortexes as you go: hitting the top center vortex will take you to the bottom center one. Keep flying straight up until you hit the center vortex again. Hit the top center vortex again, then the middle one again to solve the puzzle. This can be done the same way by traveling left or right: turn right, hit the center right vortex box, which takes you to the left center box. Hit the middle vortex again, then the right one again, then the middle one again to finish the level, or do it the exact same way but by thrusting left.
The long way involves thrusting into each center vortex, then the middle vortex box after every one. For example, fly straight up and hit the top middle vortex, which takes you to the bottom middle. Fly into the middle vortex, then the right center vortex, fly into the middle vortex again, then the bottom middle, hit the middle vortex, again, then the left middle vortex, and then the middle vortex one final time and you’re out. This can be done in any order (down/middle/up/middle/right/middle/left/middle, for instance), just as long as the center vortex is flown into every other time, and not messing with any of the vortexes in any of the corners at all.
Sectilius Demonicus: this one starts off with a sectoid latched onto your ship, so you’re going to have to kill it before proceeding. Sectoids spin around your ship at the exact same rate of speed that you move, so if you fire and see that your shot is ahead of the sectoid, let off on the controller for a split second before resuming turning. If your shot is behind the sectoid, turn around in the opposite direction to try to line up a shot, then go back to turning in the direction that it is flying.
After that, just fly continuously one way and that will solve the puzzle: hit the top vortex box, then the middle, top, then middle again to solve it, or fly straight left or right, hitting all vortexes in the same manner (by flying straight in one direction, either going left or right) to solve the puzzle.
The Darkest Star: thrust down to the A vortex (see picture), which will transport you to either the 1 or 2 vortex. Then thrust to the opposite diagonal vortex from where you started from, i. e. if you arrived at vortex 1, then you’ll thrust to vortex C, then repeat on the other side, starting at vortex B and then flying into vortex 2 to finish the puzzle.
The Flames of Perdition: first off, get as far away from the two converging Mourning Stars as possible, before they collide and take you out with their ensuing explosion (or before they collide with you and end your game that way). After that, this puzzle can be solved somewhat in the same manner as taking the long way in The Judas Paradox.
Fly to the A vortex (see photo), which takes you to the bottom left corner vortex. Fly into the middle vortex (1). Beware, this will set off the Mourning Stars, so get out of the way again. Fly into the B vortex (top, center) to reappear at the bottom middle vortex, then fly into 1 and dodge the Mourning Stars again. Then fly into the C vortex and into 1 to leave the puzzle.
The Wages of Sin: kill the sectoid first off. After that, fly into either one of the outermost vortexes, which are marked (see photo) by either A or B. Thrust into the middle vortex, then thrust into either one of the “inside” vortexes (i. e. the ones on either side of the middle vortex, marked with a 1 or 2), then thrust into the middle box to solve the puzzle. (Note: an easy way to remember this is OMIM [Outside/Middle/Inside/Middle vortexes].)
The Eyes of Baphomet: fly into one of the bottom vortexes at the left or right side of the screen, being either A or B (see photo). This will transport you to either the 1 or 2 vortexes. After that, fly into the bottom middle vortex, then hit either the 1 or 2 vortex and finish the puzzle by flying into either A or B (whichever one is left).
The Elysian Crux: fly into A vortex (see photo), then the vortex in the very center of the screen. Fly into B vortex, hit the middle vortex again, then fly into C and then the very center vortex to solve the puzzle.
The Enigma of Sorrow: fly into the A vortex (see photo, being the one lower diagonal left from the very middle vortex), then into the middle vortex. From there fly into the very bottom, diagonal left vortex of B, hit the middle vortex again, then fly into C at the very upper left to exit the puzzle.
The Leviathan Conundrum: the player starts at vortex B (see photo). Fly into the A vortex, which is the left vortex in the second from the last row on the bottom of the screen. This will take you to the very top vortex; fly into B, then into C in the very same row (being the last box on the right). Fly into B again, then D, being the right vortex in the second row from the bottom, and then thrust into B one last time to beat the game.