Parallel worlds chess

Parallel worlds chess is a three-dimensional chess variant invented by R. Wayne Schmittberger in the 1980s. The gamespace comprises three 8×8 chessboards at different levels. Each side commands two full chess armies on levels 1 and 3. Level 2 begins empty and obeys its own move rules.
Game description.
Each player controls two complete chess armies, initially on levels 1 and 3. White moves first. The first player to "capture" either enemy king wins the game.
Move rules.
On each turn, a player may move "up to three" of his (pieces and/or pawns). (On the "first" turn, however, White may move no more than "two" men.) Each man moved can move only once in the turn. The only stipulation is that none of these moves may "end" on the same level. (They may begin on the same level, however.) 
All men except kings can move straight up or down one level to an empty square. (Thus captures are not permitted when moving to a new level.)
On levels 1 and 3, moves, captures, and pawn promotions follow the same rules as in standard chess. On level 2, special rules apply:
Rules for pawns that move from level 2:
References.
Bibliography