A 1-parameter family of anisotropic models is presented. They all satisfy the Plummer law in the mass density, but have different velocity dispersions. Moreover, the stars are not confined to a particular subset of the total accessible phase space. This family is mathematically simple enough to be explored analytically in detail. The family is rich enough though to allow for a 3-parameter generalization which illustrates that even when both the mass density and the velocity dispersion profiles are required to be the same, a degeneracy in the possible distribution functions persists. The observational consequences of the degeneracy can be studied by calculating the observable radial velocity line profiles obtained with different distribution functions. It turns out that line profiles are relatively sensitive to changes in the distribution function. They therefore can be considered to be more natural observables when a determination of the distribution function is desired.