We consider a hydrodynamic-type system of balance equations for mass and momentum closed by the dynamical equation of state taking into account the effects of spatial nonlocality. We study higher symmetries and local conservation laws for this system and establish its nonintegrability for the generic values of parameters. A system of ODEs obtained from the system under study through the group theory reduction is investigated. The reduced system is shown to possess a family of the homoclinic solutions describing solitary waves of compression and rarefaction. The waves of compression are shown to be unstable. On the contrary, the waves of rarefaction are likely to be stable. Numerical simulations reveal some peculiarities of solitary waves of rarefaction, and, in particular, the recovery of their shape after the collisions Keywords nonlocal hydrodynamic-type model; integrability tests; spectral stability of soliton-like solutions; interaction of solitary waves