Axom

Axom is a 2001 game by Michel Villalonga, published by Gyptis Edition.

 The board and the pieces:

A (partial) rule description:

Michel Villalonga presents an original creation with his game Axom. The name of the game comes from Axe (axis) and Somme (sum), as the deployment axis of a stack of pieces is determined by the sum of the pieces composing it.

A stack of two or four pieces "falls" in a straight vertical or horizontal line onto the square or the three squares aligned along the axis. For instance, a stack of four splits into four parts, which either become individual stacks of one or add one piece to the stacks on which they land. A stack of three pieces, however, falls diagonally. A single piece moves one square diagonally.

The Tête (Head) is equivalent to the King in chess. Its movement is limited to one square in any of the eight possible directions, but it cannot stack with other pieces. The objective of the game is to capture the opponent's head.

The ruleset can be found here (in French).

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The type of stack movement depends of the stack's size. This reminds me of a 2003 game of mine named Abstract Chess where each stack size defines a different Chess piece. Players are able to transfer power between pieces by moving one stack piece to neighbor squares. 

the initial position is that of Chess;
there's a zrf to play abchess in Zillions

I didn't knew Axom then, and now it feels an example of independent invention even if they are not the same game. Still, I prefer mine ;-)