  We present a coherent and effective theoretical framework to systematically
construct numerically exact nonlinear solitary waves from their respective
linear limits. First, all possible linear degenerate sets are classified for a
harmonic potential using lattice planes. For a generic linear degenerate set,
distinct wave patterns are identified in the near-linear regime using a random
searching algorithm by suitably mixing the linear degenerate states, followed
by a numerical continuation in the chemical potential extending the waves into
the Thomas-Fermi regime. The method is applied to the two-dimensional,
one-component Bose-Einstein condensates, yielding a spectacular set of
waveforms. Our method opens a remarkably large program, and many more solitary
waves are expected. Finally, the method can be readily generalized to three
dimensions, and also multi-component condensates, providing a highly powerful
technique for investigating solitary waves in future works.
