The flow of incompressible fluid in elastic membrane cylidric tube is under investigation, which is modelled by an axisymmetric shell. It is considered in biological literature that provided the proper choise of the equation of state (elastic potential), the flow of blood in arteries can be modelled by such a flow. It is shown for the first time, that except the standing solitary waves in the form of the aneurizm for the rest at infinity, which take place for concrete rates of initial deformations and constant pressure, there exist two families of running solitary waves for all values of initial deformations and the speed of a fluid at infinity. This solitary waves posess speeds close to those given by the linear dispersion relation. It is shown also that mentioned solitary waves persist for the full system in a sense that they are uniformely aproximated by corrresponding solitary waves of truncated equations. Dynamical stability of standing solitary waves in the form of the aneurizm is considered.