Rösselsprung

Rösselsprung is a 1946 uncredited game, published by Hans Heidl Spiele.

Rösselsprung, meaning Knight's Move, is played on the following board:

the game can be played between two and four players

The original rules: 

• Each player receives four pieces, which are placed in the starting position — that is, on the fields marked with a circle for setup.
• With three or four players: two players each move two pieces to the left and two pieces to the right from the starting point on the game board. 
• With only two players: all four pieces of each player move within the field boundaries in opposite directions.
• Movement is made in a “knight’s move” (as in chess): that is, one square forward and two squares to the right or left, or two squares forward and one square to the right or left.
• If a piece lands on a square already occupied by another piece during its move, that piece is captured and returned to its starting position, where it must begin again. Capturing is mandatory. Failure to capture results in the moving piece being sent back to its starting position.
• Each player tries to move their four pieces to the opposite side of the board while preventing opponents from reaching their goals by capturing them.
• The winner is the first player to bring all four of their pieces into the opposite starting area.

In 1978, another game with the same name, appeared on the German market:

 

This is probably not a coincidence, given how similar the boards are:

The BGG description: The board has the shape of a cross. The middle of the cross is the common zone, while the outer parts of the board are reserved as starting/ending section for specific players. Each player starts with seven knights/horses and must try to bring them to the opposite side of the board. However, there are some instructions for that: all knights must cross the starting section before the can cross the common section and finally enter the ending section. Movements are determined with the roll of a die, where the thrown number is the number of (chess-)knight moves that 1 single knight must perform in one turn. Starting and ending sections are save, but captures are allowed in the common zone. 

This is, however, a game with random elements, since it uses a die to decide move range.