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We develop an elementary method for proving the PBW theorem for associativealgebras with an ascending filtration. The idea is roughly the following. Atfirst, we deduce a proof of the PBW property for the {\it ascending} filtration(with the filtered degree equal to the total degree in $x_i$'s) to a suitablePBW-like property for the {\it descending} filtration (with the filtered degreeequal to the power of a polynomial parameter $\hbar$, introduced to theproblem). This PBW property for the descending filtration guarantees thegenuine PBW property for the ascending filtration, for almost allspecializations of the parameter $\hbar$. At second, we develop some veryconstructive method for proving this PBW-like property for the descendingfiltration by powers of $\hbar$, emphasizing its integrability nature.  We show how the method works in three examples. As a first example, we give aproof of the classical Poincar\'{e}-Birkhoff-Witt theorem for Lie algebras. Asa second, much less trivial example, we present a new proof of a result ofEtingof and Ginzburg [EG] on PBW property of algebras with a cyclicnon-commutative potential in three variables. Finally, as a third example, wefound a criterium, for a general quadratic algebra which is thequotient-algebra of $T(V)[\hbar]$ by the two-sided ideal, generated by$(x_i\otimes x_j-x_j\otimes x_i-\hbar\phi_{ij})_{i,j}$, with $\phi_{ij}$general quadratic non-commutative polynomials, to be a PBW for genericspecialization $\hbar=a$. This result seems to be new.

The PBW property for associative algebras as an integrability condition