Three new classes of exact solution of the thermocline equations are obtained through use of an ansatz based method (Clarkson and Kruskal, 1989), which is related to the older Classical Lie Group Method used by Salmon and Hollerbach (1991) to obtain exact solutions. The newer method has not previously been applied to oceanographic problems. Our results are more general than those of Salmon and Hollerbach in two distinct ways: we obtain new classes of solution not obtainable by the older method and, in addition, we determine solutions in which the vertical temperature diffusion profile is an arbitrary function of depth. Application of the solutions to understand the control of the main thermocline is considered in a companion paper.