  Modern weather and climate prediction models are based on a system of
nonlinear partial differential equations called the primitive equations. Lie
symmetries of the primitive equations with zero external heating rate are
computed and the structure of its maximal Lie invariance algebra, which is
infinite-dimensional, is studied. The maximal Lie invariance algebra for the
case of a nonzero constant Coriolis parameter is mapped to the case of
vanishing Coriolis force. The same mapping allows one to transform the
constantly rotating primitive equations to the equations in a resting reference
frame. This mapping is used to obtain exact solutions for the rotating case
from exact solutions for the nonrotating equations. Another important result of
the paper is the computation of the complete point symmetry group of the
primitive equations using the algebraic method.
