 Timelike geodesics of an inverse-time conformal metric

In a positive-time patch, proper-time first integrals give $\dot{x}=p{t}^2$ and ${\dot{t}}^2=t^2+p^2{t}^4$. If $p\ne 0$, the unparametrized timelike geodesics are $t^2=(x-x_c{)}^2-p^{-2}$ on an appropriate branch; in the expanded quadratic form their discriminant is positive. If $p=0$, there is the separate family $x=x_c$, $t=t_{\ast }{e}^{\pm \tau }$. A description using only the nonvertical quadratic family omits genuine timelike geodesics.