  We found that the two-dimensional Schr\"odinger equation for 3 electrons in
an homogeneous magnetic field (perpendicular to the plane) and a parabolic
scalar confinement potential (frequency $\omega_0$) has exact analytical
solutions in the limit, where the expectation value of the center of mass
vector $\bf R$ is small compared with the average distance between the
electrons. These analytical solutions exist only for certain discrete values of
the effective frequency $\tilde \omega=\sqrt{\omega_o^2 + ({\omega_c \over 2}
)^2}$. Further, for finite external fields, the total angular momenta must be
M_L=3 m with m=integer, and spins have to be parallel. The analytically
solvable states are always cusp states, and take the components of higher
Landau levels into account. These special analytical solutions for 3 particles
and the previously published exact analytical solutions for 2 particles can be
written in an unified form. These formulae, when {\em ad hoc} generalized to N
coordinates, can be discussed as an ansatz for the wave function of the
N-particle system. This ansatz fulfills the following demands: it is exact for
two particles and for 3 particles in the limit of small $\bf R$ and for the
solvable external fields, and it is an eigenfuncton of the total orbital
angular momentum. The Laughlin functions are special cases of this ansatz for
infinite solvable fields and equal pair- angular- momenta.
