For a smooth, 1-periodic, positive function $c:\mathbb{R}\rightarrow (0,\infty)$ consider the following second order ODE system for the curve $\gamma(t)= (x(t),y(t),z(t)):\mathbb{R}\rightarrow \mathbb{R}^3$ (note that I will denote the derivative of $c$ by a prime and the derivatives of the curve components $x,y,z$ by dots).
\begin{align} (i) \hspace{15mm} \ddot{x}(t)+ \frac{c'(z(t))}{c(z(t))}~ \dot{x}(t)\dot{z}(t)=0 \\ (ii) \hspace{14mm} \ddot{y}(t) +\frac{c'(z(t))}{c(z(t)))}~\dot{y}(t)\dot{z}(t)=0 \\ (iii) \hspace{16mm} \ddot{z}(t)-c'(z(t)) ~ \dot{x}(t)\dot{y}(t)=0 \end{align}
My question: I'm interested in finding a function $c$ as described above such that the system has "incomplete" solutions, i.e. solutions $\gamma:\mathbb{R}\rightarrow \mathbb{R}^3$ which are not defined for all times $t\in\mathbb{R}$. Or alternatively I want to show that for all $c$ as above there are no incomplete solutions.
Background: the background to this problem is a geometric one: I have a family of metrics $\{g_c\}$ given on $\mathbb{R}^3$ (where $c$ is as described above) and I'm asking myself if all those metrics are geodasically complete. Solving the geodesic equation for those metrics gives the above system of ODEs.
The equation system has the following first integrals: $$c(z)^2\left(\dot x^2+\dot y^2\right)=I_1=const$$ $$c(z)\left(y\dot x-x\dot y\right)=I_2=const$$ If $c_{MIN}<c(z)<c_{MAX}$, then $\dot x$ and $\dot y$ are bounded: $$|\dot x|<\frac{\sqrt{I_1}}{c_{MIN}}$$ $$|\dot y|<\frac{\sqrt{I_1}}{c_{MIN}}$$ Assuming that $c'(z)$ is a bounded smooth function, $\ddot z(t)$ is also bounded. This means that there are no singular solutions (such that $x$, $y$ or $z$ reach infinity for finite $t$). Also, existence theorem guarantees existence of solution to the initial value problem in the neighborhood of every $t=t_0$, so there are no incomplete solutions.