  Let $X$ be a Hamiltonian vector field defined on a symplectic manifold
$(M,\omega)$, $g$ a nowhere vanishing smooth function defined on an open dense
subset $M^0$ of $M$. We will say that the vector field $Y = gX$ is conformally
Hamiltonian. We prove that when $X$ is complete, when $Y$ is Hamiltonian with
respect to another symplectic form $\omega_2$ defined on $M^0$, and when
another technical condition is satisfied, there exists a symplectic
diffeomorphism from $(M^0,\omega_2)$ onto an open subset of $(M,\omega)$,
equivariant with respect to the flows of the vector fields $Y$ on $M^0$ and $X$
on $M$. This result explains why the diffeomorphism of the phase space of the
Kepler problem restricted to the negative (resp. positive) values of the energy
function, onto an open subset of the cotangent bundle to a three-dimensional
sphere (resp. two-sheeted hyperboloid), discovered by Gy\"orgyi (1968) [9],
re-discovered by Ligon and Schaaf (1976) [15], whose properties were discussed
by Cushman and Duistermaat (1997) [5], is a symplectic diffeomorphism.
Infinitesimal symmetries of the Ke- pler problem are discussed, and it is shown
that their space is a Lie algebroid with zero anchor map rather than a Lie
algebra.
