I have an ODE system $$ \ddot p = \frac{ p \left( {2p - 4} \right) }{{p - 4}}{{\dot q }^2 } \\ \ddot q = \frac{{3p - 8}}{{p - 4}}\dot q \dot p $$
Short of finding closed-form expressions for $p\left( t \right)$ or $q\left( t \right)$, I have made a guess at a closed-form expression for a locus $p\left( q \right)$, namely,
$$p\left( q \right) = \frac{8}{{5 + 3\cos {\textstyle{3 \over 2}}q }}$$
(I arrived at this purely by trying to guess a function that gives a good eyeball match to a numerical solution when plotted, for initial conditions $p_0=1$, $q_0=0$, $\dot p_0=0$, $\dot q_0=1$.)
Of course, if the ODE was in a form where the derivative of $p$ was taken with respect to $q$, I could just plug my guess into the equation and see if it holds. But both $p$ and $q$ are functions of a parameter $t$. In this case, is there any way to test my guess?
Unfortunately I think that your guess is not exact (may be it is an approximate on a range). This can be verified with the method below :
The analytic solution involves complicated integrals. The first one can be expressed on a closed form involving only elementary functions, but as a big formula (not written on the page below). The next one is even more complicated and more likely cannot be expressed on a closed form. I think that one have to use numerical calculus to solve the ODE system.