  A single closed-form analytical solution of the driven nonlinear
Schr\"{o}dinger equation is developed, reproducing a large class of the
behaviors in Kerr-comb systems, including bright-solitons, dark-solitons, and a
large class of periodic wavetrains. From this analytical framework, a Kerr-comb
area theorem and a pump-detuning relation are developed, providing new insights
into soliton- and wavetrain-based combs along with concrete design guidelines
for both. This new area theorem reveals significant deviation from the
conventional soliton area theorem, which is crucial to understanding cavity
solitons in certain limits. Moreover, these closed-form solutions represent the
first step towards an analytical framework for wavetrain formation, and reveal
new parameter regimes for enhanced Kerr-comb performance.
