On the upper bound of the number of functionally independent focal quantities of the Lyapunov differential system

Denote by $N_1=2\sum\limits_{i=1}^{\ell}(m_i+1)+2$ the maximal possible number of non-zero coefficients of the Lyapunov differential system $\dot{x}= y+\sum\limits_{i=1}^{\ell}P_{m_i}(x,y)$, $\dot{y}= -x+\sum\limits_{i=1}^{\ell}Q_{m_i}(x,y)$, where $P_{m_i}$ and $Q_{m_i}$ are homogeneous polynomials of degree $m_i$ with respect to $x$ and $y$, and $1<m_1<m_2<...<m_{\ell}$ $(\ell<\infty)$. Then the upper bound of functionally independent focal quantities in the center and focus problem of considered system does not exceed $N_1-1$.