A new type of non-uniqueness for the scalar geodetic boundary value problem

The problem of non-uniqueness of the scalar geodetic boundary value problem has been treated in recent geodetic literature. It has been pointed out th at it is partly an open problem. It is demonstrated that in its nonlinear f orm, the problem exhibits many more features of ill-posedness than were kno wn previously. Moreover, it is shown that the non-uniqueness appears mainly in the purely gravitational version of the problem, which is usually consi dered for mathematical analysis. It is also shown that in a more general ve rsion of the problem, which does not assume the rotational potential of the Earth to be known at the boundary, there is at least one special case of n on-uniqueness, belonging approximately to a spheroidal planet of 1/116 flat tening.