Integrating factors and integrability of an ODE system

Question

The following argument is from a paper about the Bendixson-Dulac Theorem.

Consider a smooth differential equation on the plane $$ x'=g(x,y),\quad y'=h(x,y). $$ Suppose there exists a function $D(x,y)$ such that $$ (Dg)_x+(Dh)_y=0. $$ Then $D$ is an integrating factor and the system is integrable.

A quick search for "integrable system" on Google returns results not satisfying.

Could anyone explain what the last sentence in the argument above means?

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In the paper quoted above,

$$ g(x,y)=ax+bx^2+cxy,\quad h(x,y)=dy+exy+fy^2 $$

and $D(x,y)=x^ry^s$ for some $r,s$.

Answer 1

$X=g\partial_x + h\partial_y$ is the vector field whose flow lines are wanted.
$\omega=hdx - gdy$ is a 1-form with kernel the span of $X$.
Also $D\omega$ has kernel the span of $X$ for any function $D$ which does not vanish anywhere. If $d(D\omega)=0$ (this is your condition) then $D\omega$ is a closed 1-form, thus exact on simply connected sets. So $D\omega = dF$ for a function $F$ which can easily be computed by line integrals. Thus the wanted flow lines are contained in the level sets of $F$.
Finally, the time dependence of the flow has to be computed extra.