Analytic axisymmetric galaxy models with three integrals of motion

We investigate a family of inhomogeneous axisymmetric mass models with a simple gravitational potential originally introduced by Kuzmin, and subsequently studied by Kuzmin and Kutuzov. Properties of the mass models are presented, for both the oblate and the prolate sequences. Their spherical limit is Henon's isochrone. The density distribution ρ(π, z) of the models is stratified approximately on spheroids and can be written explicitly as ρ(π,ψ), where ψ is the gravitational potential and (π, z) are cylindrical coordinates. This fact makes it possible to use a standard inversion technique in order to obtain-in closed form-the unique distribution function F(E, L^2^_z_) that depends only on the two classical integrals of motion (the energy E and the component L_z_ of the angular momentum parallel to the symmetry axis) and that is consistent with the density. This F is nonnegative for all oblate models in the sequence, and also for the prolate models with central axis ratio smaller than 1.35. The resulting intrinsic velocity dispersions are given explicitly. The potential of all these models is of Stackel form, so that all orbits in them have an exact third integral of motion I_3_, which can be regarded as a generalization of the total angular momentum integral of the spherical limit. We present a new method for the analytical construction of distribution functions F(E, L^2^_z_, I_3_), and apply it to these mass models. We write F as the sum of two parts: F(E, L^2^_z_, I_3_) = F_1_(E, L^2^_z_) + F_2_(E, L^2^_z_, I_3_), where F_2_ is a power series in E, L^2^_z_, and I_3_. We choose a simple form for F_2_, compute the resulting density and subtract it from the given ρ. The remaining density is then reproduced by F_1_, which is obtained by the standard inversion technique. This produces-for the first time-exact analytic distribution functions for realistic axisymmetric models that depend on all three integrals of motion. Kinematic properties of the models are presented, and observables are calculated.