Isometric immersions of rotation metrics on a sphere into $E^4$ in the form of surfaces of revolution

In this paper, under investigation is the problem of immersion of a metric on $S^2$, which has the form $ds^2=g(r) (dr^2+r^2d\phi^2)$ in a certain stereographic projection, into $Е^4$ in the form of a surface of revolution with a pole. Denote the poles as $О_1$ and $O_2$. It is established that there exists an immersion in the class $C^1(S^2) \cap C^2(S^2\setminus (O_1\cup O_2))$. Some necessary and sufficient conditions are given for an immersion in the class $C^2(S^2)$. The deformability of the resulting surfaces in the same class is proved.