The governing partial differential equations for static deformations of homogeneous isotropic compressible hyperelastic materials (sometimes referred to simply as perfectly elastic materials) are highly nonlinear and consequently only a few exact solutions are known. For these materials, only one general solution is known which is for plane deformations and is applicable to the so-called harmonic materials originally introduced by John. In this paper we extend this result to axially symmetric deformations of perfectly elastic harmonic materials. The results presented hinge on a reformulation of the equilibrium equations and a similar procedure can be exploited to derive the known solution due to John. It is shown that axially symmetric deformations of a harmonic material can be reduced to two linear equations whose coefficients involve the partial derivatives of an arbitrary harmonic function ω(R, Z). For any harmonic material, this linear system admits a simple general solution for the two special cases ω = ω(R) and ω = ω(Z). For the ‘linear-elastic’ strain-energy function, the linear system is shown to admit a general solution for the two harmonic functions which, in spherical polar coordinates, arise from the assumptions of spherical and radial symmetry.