  A continuous family of singular solitary waves exists in a prototypical
system with intensity-dependent dispersion. The family has a cusped soliton as
the limiting lowest energy state and is formed by the solitary waves with
bell-shaped heads of different lengths. We show that this family can be
obtained variationally by minimization of mass at fixed energy and fixed length
of the bell-shaped head. We develop a weak formulation for the singular
solitary waves and prove that they are stable under perturbations which do not
change the length of the bell-shaped head. Numerical simulations confirm the
stability of the singular solitary waves.
