  For a system of ordinary differential equations (ODEs) or, more generally, an
involutive distribution of vector fields, the problem of its integration is
considered. Among the many approaches to this problem, solvable structures
provide a systematic procedure of integration via Pfaffian equations that are
integrable by quadratures. In this paper structures more general than solvable
structures (named cinf-structures) are considered. The symmetry condition in
the concept of solvable structure is weakened for cinf-structures by requiring
their vector fields be just cinf-symmetries. For cinf-structures there is also
an integration procedure, but the corresponding Pfaffian equations, although
completely integrable, are not necessarily integrable by quadratures. The
well-known result on the relationship between integrating factors and Lie point
symmetries for first-order ODEs is generalized for cinf-structures and
involutive distributions of arbitrary corank by introducing symmetrizing
factors. The role of these symmetrizing factors on the integrability by
quadratures of the Pfaffian equations associated with the \cinf-structure is
also established. Some examples that show how these objects and results can be
applied in practice are also presented.
