  In this paper, we present a systematic study of the force-free field equation
for simple axisymmetric configurations in spherical geometry and apply it to
the solar active regions. The condition of separability of solutions in the
radial and angular variables leads to two classes of solutions: linear and
nonlinear force-free fields. We have studied these linear solutions and
extended the nonlinear solutions for the radial power law index to the
irreducible rational form $n= p/q$, which is allowed for all cases of odd $p$
and cases of $q>p$ for even $p$, where the poloidal flux $\psi\propto1/r^n$ and
field $\mathbf{B}\propto 1/r^{n+2}$). We apply these solutions to simulate
photospheric vector magnetograms obtained using the spectropolarimeter on board
Hinode. The effectiveness of our search strategy is first demonstrated on test
inputs of dipolar, axisymmetric,and non axisymmetric linear force-free fields.
Using the best-fit to these magnetograms, we build three-dimensional
axisymmetric field configurations and calculate the energy and relative
helicity with two independent methods, which are in agreement. We have analyzed
five magnetograms for AR 10930 spanning a period of three days during which two
X-class flares occurred which allowed us to find the free energy and relative
helicity of the active region before and after the flare; our analysis
indicates a peak in these quantities before the flare events which is
consistent with the results mentioned in literature. We also analyzed
single-polarity regions AR 10923 and 10933, which showed very good fits with
potential fields. This method can provide useful reconstruction of the
nonlinear force-free (NLFF) fields as well as reasonably good input fields for
other numerical techniques.
