Group foliations, invariant solutions, and conservation laws of the geopotential forecast equation

Despite the large number of publications on symmetry analysis of the geopotential forecast equation, its group foliations laws have not been considered previously. The present publication aims to address this shortcoming. First, group foliations are constructed for the equation, and based on them, invariant solutions are derived, some of which generalize previously known exact solutions. There is also a discussion of the pros and cons of the group foliation approach. In addition, the rest of the paper is dedicated to a comprehensive discussion of conservation laws. All possible second-order conservation laws of the geopotential forecast equation are obtained through direct calculations, and a number of higher-order conservation laws are derived using the known symmetries of the equation.