I have a problem in determining the number of integration constants of a system of coupled differential equations
$$ \phi''(r) + 4A'(r)\phi'(r) = \frac{\partial V(\phi)}{\partial \phi}\\ A''(r) = -\frac{2}{3}\phi'(r)\\ \left[A'(r)\right]^2 = -\frac{1}{3}V(\phi) + \frac{1}{6}\left[\phi'(r)\right]^2 $$
where $\phi(r)$ and $A(r)$ depend on $r\in \mathbb{R}$ and $V(\phi)$ is a generic function of $\phi(r)$.
The authors of this paper https://arxiv.org/abs/hep-th/9909134 (see Eq.(6) and discussion in page 7) state the integration constants are three. I am not able to see how obvious is this counting.
Question: How do I count the integration constants of these system of coupled non-linear differential equations?
Notice that this system can be equivalenty written in terms of a "superpotential" $W(\phi)$ from which should be clear the integration constants are three
$$ V(\phi) = \frac{1}{8}\left[\frac{\partial W(\phi)}{\partial \phi}\right]^2 - \frac{1}{3} W(\phi)^2\\ \phi' = \frac{1}{2}\frac{\partial W(\phi)}{\partial \phi}\\ A' = -\frac{1}{3}W(\phi) $$
In particular, one integration constant enters in $W(\phi)$.
If the implied functions $$r\mapsto\phi(r),\quad r\mapsto A(r), \quad u\mapsto V(u)$$ all are unknown functions then the given system is not a system of coupled ODEs, nor can it be transformed to a PDE with independent variables $r$ and $u$. It is a problem of its own kind. You cannot expect that there is a general theory about such problems. Maybe the general solution of your problem contains $r\geq0$ free constants, maybe even an arbitrary function; who knows.