# Does every ODE have a first integral?

This question might have an obvious answer but it eludes me. It is equivalent to asking whether every ODE can be solved in quadratures. For example, a first order ODE can be reduced to quadratures if we can solve for the integration factor $mu$. This requires solving a first order partial differential equation, and as far as I know there are no general existence theorems on these.

A possible equivalent question would be, given a function $x=f(t,c)$, can we find two functions $F(x,t)$ and $G(c)$, such that $F(f(t,c),t)=G(c)$?

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If our ode $\dot\gamma(t)=X_{\gamma(t)}$ is associated to a nonsingular vector field $X$ on a smooth $n\textrm{-dimensional}$ manifold $M$ then any point has a coordinate system $(x_1,\ldots,x_n)$ such that $X=\frac{\partial}{\partial x_1}.$
Consequently $x_2,\ldots,x_n$ is system of $n-1$ independent first integral of the ode $\dot\gamma=X_{\gamma(t)}.$