Vector field on the unit sphere

Question

Let $\vec\psi$ be the vector field defined on the unit sphere $S^2$ in the basis $(\vec e_r,\vec e_\theta,\vec e_\varphi)$ of spherical coordinates by: $$\vec\psi(r=1,\theta,\varphi)=(0,\sin\varphi,0)$$ I want to determine the integral curves of $\vec\psi$. So I solve the system $$\begin{cases}\dfrac{dr}{dt}=0\\ \dfrac{d\theta}{dt}=\sin \varphi \\ \dfrac{d\varphi}{dt}=0 \end{cases}$$ and the integral curves are given by: $r(t)=c_1$, $\varphi(t)=c_2$ and $\theta(t)=\sin(c_2)t+c_3$ where $c_1,c_2,c_3$ are any constant numbers. Is this correct and is there any physical phenomena that can be described by such a vector field ?