On a three-dimensional Riccati differential equation and its symmetries

A three-dimensional Riccati differential equation of complex quaternion-valued functions is studied. Many properties similar to those of the ordinary differential Riccati equation such that linearization and Picard theorem are obtained. Lie point symmetries of the quaternionic Riccati equation are calculated as well as the form of the associated three-dimensional potential of the Schr\"odinger equation. Using symmetry reductions and relations between the three-dimensional Riccati and the Schr\"odinger equation, examples are given to obtain solutions of both equations.