Fermat certificates for every homogeneous odd source

For each $q\in\{1,2\}$ and every homogeneous polynomial $A\in\mathbb C[X_0,X_1,X_2,Y_0,Y_1,Y_2]$ of degree $6\cdot q$ satisfying $A(X,-Y)=-A(X,Y)$, the Fermat sextic $F=\sum_{i=0}^{2}(X_i^6+Y_i^6)$ admits a Diamond certificate: a rational vector field $v$ homogeneous of degree $6\cdot q-5$, with common denominator relatively prime to $F$, satisfying $F^{q+1}\mid A-q\cdot dF(v)+F\cdot\operatorname{div}(v)$. Divisibility is in the localization with denominators relatively prime to $F$; there is no bound on denominator degree.