On solutions of the two-point boundary problem for one non-autonomous differential system with a quadratic at phase variables right-hand side

In the paper we consider the system $\dot{x}=ax+by+a_{20}x^2+a_{11}xy+a_{02}y^2$, $\dot{y}=-bx+ay+b_{20}x^2+b_{11}xy+b_{02}y^2$, where $a_{ij}=a_{ij}(t)$, $b_{ij}=b_{ij}(t)$ are the continued functions; $a$ and $b$ are the constants. For this system we established conditions under which this system has a linear Mironenko reflecting function and therefore a linear mapping in period $[-\omega; \omega]$. The obtained conditions allow us point out the initial data of the solutions of the two-point boundary task $\Phi(x(\omega), y(\omega), x(-\omega), y(-\omega))=0$ and therefore, the initial data of the $2\omega$-periodic solutions of the system (1) in the case when its coefficients are $2\omega$ periodic continued functions.