  The notion of singular reduction modules, i.e., of singular modules of
nonclassical (conditional) symmetry, of differential equations is introduced.
It is shown that the derivation of nonclassical symmetries for differential
equations can be improved by an in-depth prior study of the associated singular
modules of vector fields. The form of differential functions and differential
equations possessing parameterized families of singular modules is described up
to point transformations. Singular cases of finding reduction modules are
related to lowering the order of the corresponding reduced equations. As
examples, singular reduction modules of evolution equations and second-order
quasi-linear equations are studied. Reductions of differential equations to
algebraic equations and to first-order ordinary differential equations are
considered in detail within the framework proposed and are related to previous
no-go results on nonclassical symmetries.
