# Potential function for an exact closed differential form and a solution of a corresponding linear first order homogeneous differential equation

A linear differential form $\sum_{i}\mathcal{E}_{i}(q)\, dq_{i}$ is an exact differential if the conditions $\partial\mathcal{E}_{i}(q)/\partial q_{j}=$ $\partial\mathcal{E}_{j}(q)/\partial q_{i}$ are met for any $i, j$.

Corresponding closed exact differential form is equivalent to a linear first order homogeneous differential equation: $$\sum_{i}\mathcal{E}_{i}(q)\frac{dq_{i}}{d\tau}=0;$$ for which, in case when the potential function $E(q)$ for the corresponding form, such that: $$\mathcal{E}_{i}(q)=\partial E(q)/\partial q_{i};$$ is known, the general one-parametric solution is simply: $$E(q_{1},...q_{n})=Const;$$

What can we say about the solution of this differential equation when the explicit potential function is not known, can not be given in a closed explicit form (does not exist)?
Can we still expect that there exist some implicit form of a general solution, as a one-parametric family of relations between variables $q_{i}$?

If so, than how to find / build those solutions?

I appreciate your suggestions, specific references.