  We address dynamics of Bose-Einstein condensates (BECs) loaded into a
one-dimensional four-color optical lattice (FOL) potential with commensurate
wavelengths and tunable intensities. This configuration lends system-specific
symmetry properties. The analysis identifies specific multi-parameter forms of
the FOL potential which admits exact solitary-wave solutions. This newly found
class of potentials includes more particular species, such as frustrated
double-well superlattices, and bi-chromatic and three-color lattices, which are
subject to respective symmetry constraints. Our exact solutions provide options
for controllable positioning of density maxima of the localized patterns, and
tunable Anderson-like localization in the frustrated potential. A numerical
analysis is performed to establish dynamical stability and structural stability
of the obtained solutions, which makes them relevant for experimental
realization. The newly found solutions offer applications to the design of
schemes for quantum simulations and processing quantum information.
