This seems indeed equivalent to the classical question treated by Darboux, Painleve, Poincare and others. Not surprisingly it can have several different formulations.

Classical formulation is the following: Given a polynomial equation
$f(x,y)dx+g(x,y)dy=0$, to find with finitely many operations an integrating factor, or to prove it does not exist. The next question is "what kind of integrating factor"? The simplest case is a rational integrating factor (though more general factors were considered in the classical papers). If we look for a rational integrating factor of degree at most $n$, we can just write it with undetermined coefficients, and the question is reduced to solving a finite system of algebraic equations. There is an algorithm of doing this, if the original equation is over rational (or algebraic) numbers. So the real question is: can we bound the degree of the factor?

Then the question is "bound in terms of what"? A relatively recent breakthrough
in this problem is the counterexample: one cannot bound the degree of the factor in terms of degrees of $f,g$, and the types of singularities: there is a one-parametric family of equations, with singularities of fixed type (that is the type is independent of parameter) such that
for a countable dense set of values of parameter it has integrating factors, but their degrees tend to infinity.

MR1914932

Lins Neto, Alcides,
Some examples for the Poincaré and Painlevé problems.
Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 2, 231–266.

This paper also contains a survey of previous results.

However it remains a possibility that the degree of the integrating factor can be estimated in terms of some arithmetic properties of the coefficients.

Here is a survey of the topic:
MR2166493
Llibre, Jaume
Integrability of polynomial differential systems. Handbook of differential equations, 437–532, Elsevier/North-Holland, Amsterdam, 2004.

All this was about the global aspect of the problem (as I said it has many formulations). The local aspect is usually called the Center-Focus Problem:
for the same kind of system, to determine whether an equilibrium point is a center or a focus. There is an enormous literature on this, and this problem is much better understood. Just look at the
"center-focus problem" as a keyword.