Commuting Krichever–Novikov differential operators with polynomial coefficients

Under study are some commuting rank 2 differential operators with polynomial coefficients. We prove that, for every spectral curve of the form $w^2=z^3+c_2z^2+c_1z+c_0$ with arbitrary coefficients $c_i$, there exist commuting nonselfadjoint operators of orders 4 and 6 with polynomial coefficients of arbitrary degree.