  A more accurate analytical solution of the {\sl vertical-current
approximation nonlinear force-free field (VCA3-NLFFF)} model is presented that
includes besides the radial $(B_r)$ and the azimuthal $(B_\varphi)$ magnetic
field components a poloidal component $(B_{\theta} \neq 0)$ also. This new
analytical solution is of second-order accuracy in the divergence-freeness
condition, and of third-order accuracy in the force-freeness condition. We
re-analyze the sample of 173 GOES M- and X-class flares observed with the {\sl
Atmospheric Imaging Assembly (AIA)} and {\sl Helioseismic and Magnetic Imager
(HMI)} onboard the {\sl Solar Dynamics Observatory (SDO)}. The new code
reproduces helically twisted loops with a low winding number below the kink
instability consistently, avoiding unstable, highly-twisted structures of the
Gold-Hoyle flux rope type. The magnetic energies agree within
$E_{VCA3}/E_W=0.99\pm0.21$ with the Wiegelmann (W-NLFFF) code. The time
evolution of the magnetic field reveals multiple, intermittent energy build-up
and releases in most flares, contradicting both the Rosner-Vaiana model (with
gradual energy storage in the corona) and the principle of time scale
separation ($\tau_{flare} \ll \tau_{storage}$) postulated in self-organized
criticality models. The mean dissipated flare energy is found to amount to
$7\%\pm3\%$ of the potential energy, or $60\%\pm26\%$ of the free energy, a
result that can be used for predicting flare magnitudes based on the potential
field of active regions.
