As a step toward developing a general method for determining the underlying geometric structure of two time-scale optimally controlled nonlinear systems, we define a degenerate class of two time-scale optimal control problems, called completely hypersensitive problems, and propose an indirect solution method for this class of problems. The method uses a dichotomic basis to split the Hamiltonian vector field into its stable and unstable components. An accurate approximation to the optimal solution is constructed by matching the initial and terminal boundary-layer segments with the equilibrium solution. A variation of the method for the case of an approximate dichotomic basis is also developed and is applied to a nonlinear spring-mass problem. The challenging problem of determining a dichotomic basis or a sufficiently accurate approximation to one is discussed only briefly, but some potential solutions are identified.