For two-dimensional autonomous analytic dynamical systems it is proved the existence of "canonical" and "quasicanonical" integrals, the important special case of which are Darbouxian integrals and their generalizations. Influence of limit sets on singularity of the first integrals and the integrating factors is studied. Erugin's problem of the existence polynomial vector fields with center and limit cycles is solved. The counterexample to K. S. Sibirsky's hypothesis on everywhere dense set of algebraic equations with Darbouxian integral in the set of equations with the center is constructed.
Main publications:
Dolov M. V. Ob algebraicheskikh predelnykh tsiklakh polinomialnykh vektornykh polei na ploskosti // Differentsialnye uravneniya. 2001. T. 37. # 9. C. 1155–1160.
Dolov M. V., Chistyakova S. A. O strukture obschego resheniya i integriruyuschego mnozhitelya v okrestnosti prostoi osoboi tochki // Differentsialnye uravneniya. 2001. T. 37. # 5. C. 710–713.
Dolov M. V. Integriruyuschii mnozhitel v okrestnosti uzla // Differentsialnye uravneniya. 1997. T. 33. # 2. C. 158–160.
Dolov M. V., Chistyakova S. A. Algebraicheskie differentsialnye uravneniya s integriruyuschim mnozhitelem tipa Darbu // Differentsialnye uravneniya. 1997. T. 33. # 5. C. 618–622.
Dolov M. V., Kruglov E. V. O chisle polualgebraicheskikh chastnykh integralov odnogo klassa dinamicheskikh sistem s tsilindricheskim fazovym prostranstvom // Differentsialnye uravneniya. 1995. T. 31. # 6. C. 949–954.
