Surfaces specified by integrable systems of partial differential equations determined by structure equations and Lax pair

A system of evolution equations can be developed from the structure equations for a submanifold embedded in a three-dimensional space. It is seen how these same equations can be obtained from a generalized matrix Lax pair provided a single constraint equation is imposed. This can be done in Euclidean space as well as in Minkowski space. The integrable systems which result from this process can be thought of as generalizing the SO(3)$SO\left(3\right)$ and SO(2,1)$SO(2,1)$ Lax pairs which have been studied previously.