The Projective Andoyer transformation and the connection between the 4-D isotropic oscillator and Kepler systems

Extending to 4 degrees of freedom a symplectomorphism used in attitude dynamics it is shown in a direct way the connection between the 4-D isotropic harmonic oscillator and the 3-D Kepler systems. This approach made transparent that only when we refer to rectilinear solutions, the {\sl bilinear relation} defining the KS transformation is needed.