  Boussinesq systems of nonlinear partial differential equations are
fundamental equations in geophysical fluid dynamics. In this paper, we use
asymmetric ideas and moving frames to solve the two-dimensional Boussinesq
equations with partial viscosity terms studied by Chae ({\it Adv. Math.} {\bf
203} (2006), 497-513) and the three-dimensional stratified rotating Boussinesq
equations studied by Hsia, Ma and Wang ({\it J. Math. Phys.} {\bf 48} (2007),
no. 6, 06560). We obtain new families of explicit exact solutions with multiple
parameter functions. Many of them are the periodic, quasi-periodic, aperiodic
solutions that may have practical significance. By Fourier expansion and some
of our solutions, one can obtain discontinuous solutions. In addition, Lie
point symmetries are used to simplify our arguments.
