On algebraic integrals of the Hill problem and restricted circular planar three-body problem on a level of energy

It is established that the restricted circular planar three-body problem (RCPTBP) [1], [15], [5] admits a nonconstant algebraic integral on a level of energy only in cases when it can be reduced to the Kepler problem. The Hill problem [1], [7], [5] is the limit case of the RCPTBP if by analogy with the Moon-Earth-Sun system we put the mass of the Sun and the distance between the Sun and the Earth to be infinitely large. It is established that the Hill problem also does not admit a non-constant algebraic integral on any level of energy. The proof is based on the Husson method [8], [2], improved by the author [21], [22]. At the end of the proof we expand the result of J. Liouville [13] that the integral $\int f(z)e^z$ for $f$ algebraic in $z$ is not generally an algebraic function times the exponent function.