## Coefficient conditions for the topological selection of quadratic systems of Darboux type.(Russian)Zbl 0714.34054

For the system $dx/dt=gx+hy+mx^ 2+2nxy+py^ 2,\quad dy/dt=kx+\ell y+qx^ 2+2rxy+sy^ 2$ with real coefficients K. S. Sibirskij [Introduction to the algebraic theory of invariants of differential equations. (Russian) (Kishinev 1982; Zbl 0559.34046)] presented a minimal polynomial basis of centroaffine comitants. Using polynomials of the elements of this basis we find centroaffine-invariant coefficient conditions for this system to be a Darboux system of the type $dx/dt=x+P_ 2(x,y),\quad dy/dt=y+Q_ 2(x,y)$ or $dx/dt=gx+hy+xR(x,y),\quad dy/dt=kx+\ell y+yR(x,y)$ where R(x,y)$$\not\equiv 0$$ denotes a linear form in x and y.

### MSC:

 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.

Zbl 0559.34046
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