The authors construct polynomial systems of degree \(n>3\), which for even \(n\) have a week focus of order \(n^2-1\) and for odd \(n\) have a week focus of order \((n^2-1)/2\) at the origin. The systems are from the family of the form \[ \dot x=-y(1-f_\alpha(x,y))+P(x,y), \;\dot y=x(1-f_\alpha(x,y))+Q(x,y), \] where \( f_\alpha(x,y) \) is a real homogeneous polynomial of degree \(n-1\) depending on the real parameter \(\alpha\), \(P\) and \(Q\) are homogeneous polynomials of degree \(n\) with small real coefficients. The obtained result improves previously known estimates.