  We address the nonlinear Schrodinger equation with intensity-dependent
dispersion which was recently proposed in the context of nonlinear optical
systems. Contrary to the previous findings, we prove that no solitary wave
solutions exist if the sign of the intensity-dependent dispersion coincides
with the sign of the constant dispersion, whereas a continuous family of such
solutions exists in the case of the opposite signs. The family includes two
particular solutions, namely cusped and bell-shaped solitons, where the former
represents the lowest energy state in the family and the latter is a limit of
solitary waves in a regularized system. We further analyze the delicate
analytical properties of these solitary waves such as the asymptotic behavior
near singularities, the spectral stability, and the convergence of the
fixed-point iterations near such solutions. The analytical theory is
corroborated by means of numerical approximations.
