  Any singular level of a completely integrable system (c.i.s.) with
non-degenerate singularities has a singular affine structure. We shall show how
to construct a simple c.i.s. around the level, having the above affine
structure. The cotangent budle of the desingularised level is used to perform
the construction, and the c.i.s. obtained looks like the simplest one
associated to the affine structure. This method of construction is used to
provide several examples of c.i.s. with different kinds of non-degenerate
singularities.
