  We study a specific class of deformations of curve singularities: the case
when the singular point splits to several ones, such that the total $\delta$
invariant is preserved. These are also known as equi-normalizable or
equi-generic deformations. We restrict primarily to the deformations of
singularities with smooth branches.
  A natural invariant of the singular type is introduced: the dual graph. It
imposes severe restrictions on the possible collisions/deformations. And allows
to prove some bounds on the variation of classical invariants in
equi-normalizable families.
  We consider in details deformations of ordinary multiple points, the
deformations of a singularity into the collections of ordinary multiple points
and deformations of the type $x^p+y^{pk}$ into the collections of $A_k$'s.
