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

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