Integrability of polynomial vector fields and a dual problemPetek, Tatjana (Avtor) Romanovski, Valery (Avtor) integrabilnostpolinomialni sisteminvariantnopolynomial system of ODEsintegrabilityfirst order linear PDEmultigraded ringsinvariantsWe investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions annihilated by differential operators and explore the properties of the integrability variety in relation to the invariants of the associated Lie group. Our study extends to differential operators on affine algebraic varieties, highlighting the intrinsic connection between these operators and local analytic first integrals. To illustrate the duality, the case of quadratic vector fields is considered in detail.Springer Nature20262026-05-15 09:06:59Članek v reviji98091UDK: 51COBISS_ID: 278060035DOI: 10.1007/s12346-026-01484-2ISSN pri članku: 1662-3592sl© The Author(s) 2026