A cubic system with eight small-amplitude limit cycles

In E.M. James and N.G. Lloyd's paper A Cubic System with Eight Small-Amplitude Limit Cycles [1], a set of conditions is given that ensures the origin to be a fine focus of order eight and eight limit cycles to bifurcate from the origin by perturbing parameters. We find that one of the conditions, a₉ = σ_*a₇, where 666/97 < σ_* < 103/15, can be weakened as a₉ = σ_*a₇ or a₉ = σ₁a₇, where 283/125 < σ₁ < 284/125. In [1], deriving above conditions is reduced to finding the real solutions of a system of some algebraic equations and inequalities. When verifying these conditions by solving this system in a different ordering, we find another real solution to the system, which is leading to above improvement of the conditions.