The focus of this work is the development of general techniques of stochastic averaging of randomly-perturbed four-dimensional integrable Hamiltonian systems. The integrable system here has certain nontrivial (yet generic) types of fixed points. Stochastic averaging makes use of these integrable structures to identify a reduced diffusive model on a space which encodes the structure of the fixed points and can have dimensional singularities. At these singularities, glueing conditions will be derived, these glueing conditions completing the specification of the dynamics of the reduced model. Qualitative dependence of statistical measures of the reduced system upon various coefficients can be studied by extensions of known techniques such as stochastic bifurcation theory.