Applying our theory to autonomous analytic differential systems, we obtain some conditions on the existence of limit cycles and integrability.
For polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues, we prove that there always exists a positive number N such that if its first N averaging functions vanish, then all averaging functions vanish, and consequently there exists a neighborhood of the origin filled with periodic orbits. Consequently if all averaging functions vanish, the origin is a center for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615006117&_mathId=si1.gif&_user=111111111&_pii=S0022039615006117&_rdoc=1&_issn=00220396&md5=694d7d2b4f819ae318f1bbabbafc306e" title="Click to view the MathML source">n=2class="mathContainer hidden">class="mathCode">.
Furthermore, in a punctured neighborhood of the origin, the system is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615006117&_mathId=si132.gif&_user=111111111&_pii=S0022039615006117&_rdoc=1&_issn=00220396&md5=548c600cb4091af31da8cfc2e3dcd60c" title="Click to view the MathML source">C∞class="mathContainer hidden">class="mathCode"> completely integrable for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615006117&_mathId=si131.gif&_user=111111111&_pii=S0022039615006117&_rdoc=1&_issn=00220396&md5=4f41e13610ea35137a3c05ba7e7eee04" title="Click to view the MathML source">n>2class="mathContainer hidden">class="mathCode"> provided that each periodic orbit has a trivial holonomy.
Finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities.