On the properties of the solutions of special differential equations of the third order

where p = p(x) ~ O, r -- r(x) ~ 0 are continuous functions of x C (-- ~, ~) and r(x) ~ 0 holds only for the isolated points. In this paper we derive some properties of the solutions of the differential equations (a) and (b), we prove that (b) is the so called quasi-adjoint differential equation of the differential equation (a) and we derive some relations between the solutions of the equations (a) and (b). We introduce the bands of solutions of (a) and derive the oscillatory properties of the solutions of (a) and (b), which follow from the properties of the bands. Then we prove the existence of the solutions without zero points in (--~, ~).