Maybe someone can verify my answers. The problem is as follows:
Consider a line $l$ in $\mathbb{R}^3$ and rotate it around the $Oz$ axis. Denote by $A$ the set of all such obtained surfaces.
Question 1: Write their parametrizations.
My answer: Let $S \in A$. Then there exists a line $\phi:\mathbb{R}\rightarrow\mathbb{R}^3:t\mapsto \phi(t)=p_0+tv$ and a matrix $R(\theta)$ corresponding to the rotation around the $Oz$ axis by an angle $0\leq \theta <2\pi$ such that $S$ is parametrized as $\psi: [0,\theta]\times \mathbb{R}\rightarrow S:(\alpha,t)\mapsto R(\alpha)\phi(t).$
Queston 2: Describe as many geodesics as you can for each surface.
My partial answer: I started with taking an $S$ as above and defining $\gamma(t)=\psi(\alpha,t)/||v||$, where the vector $v$ is a "direction" of the line $l$. Since \gamma''(t)=(0,0,0), $\gamma$ is a geodesic on $S$.
How can I find many others?