Desingularization of finitely smooth vector fields on the plane

The paper is devoted to desingularization of finitely smooth vector fields on the plane. Let the difference of germs of a finitely smooth vector field and an analytic vector field be sufficiently flat at a common singular point. Then these vector fields share similar behavior under desingularization. In particular, the nice desingularizations of both vector fields are achieved in the same number of steps. Moreover, for any analytic vector field with a singular point of multiplicity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu 0"> <mml:semantics> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, nice desingularization of the finitely smooth vector field is achieved after <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu 0 plus 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu _0+2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> steps. In addition, a description is presented for topologically sufficient jets of finitely smooth vector fields at nonmonodromic singular points.