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Solutions for all Adler-Bobenko-Suris equations excluding Q4 and severallattice Boussinesq-type equations are reconsidered by employing the Cauchymatrix approach. Through introducing a ``fake'' nonautonomous plane wavefactor, we derive soliton solutions, oscillatory solutions, andsemi-oscillatory solutions, for the target lattice equations. Unlike theconventional soliton solutions, the oscillatory solutions take constant valueson all elementary quadrilaterals on Z^2, which demonstrate periodic structure.

Revisit to solutions for Adler-Bobenko-Suris lattice equations and lattice Boussinesq-type equations