I am working in a Differential Geometry problem which involves finding the explicit flow of the following vector field $$X=\frac{\partial}{\partial y}-z\log z\tan y\frac{\partial}{\partial z}.$$ Its domain is the open set $W\subset\mathbb R^3$, where $$W=\{(x,y,z)\in\mathbb R^3\,|\,-\pi/2<y<\pi/2,\,z>0\}.$$ By applying the separation of variables method, we find that $$\gamma(t)=\big(x,y+t,e^{\cos(y+t)}\big)$$ is an integral curve of $X$ that passes through the point $(x,y,e^{\cos(y)})$ at $t=0$. I would like to find, however, integral curves that pass through $(x,y,z)$, for any arbitrary $z>0$.
Thanks in advance for your time.
You're almost there. You just need to solve the IVP $\dot{z}(t)=-z(t)\log z(t)\,\tan(y_0+t)$, with $z(0)=z_0$. The solution is \begin{align} z(t)&=e^{\frac{\log z_0}{\cos y_0}\cos(y_0+t)}, \end{align} defined for $t\in\Bbb{R}$ such that $y_0+t\in \left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. So, really, all you were missing was a constant of integration, $A$, in $z(t)=e^{A\cos(y(t))}$.
So, the flow is the map $\Phi:\Omega\to W$, \begin{align} \Phi_t(x,y,z)=\left(x,y+t,e^{\frac{\log z}{\cos y}\cos(y+t)}\right), \end{align} where $\Omega=\{(t,x,y,z)\in\Bbb{R}^4\,:\, \text{$t+y\in (-\pi/2,\pi/2)$ and $y\in (-\pi/2,\pi/2)$ and $z>0$}\}$. As expected, the flow is incomplete.