Generalized skew-derivations acting as commuting maps on prime rings

Suppose R is a non-commutative prime ring with char\((R)\ne 2\). Suppose that \(\kappa \left( x_{1}, \ldots , x_{n}\right) \) is a noncentral multilinear polynomial over C, \(S=\{\kappa (\wp _1,\ldots ,\wp _n) \mid \wp _1,\ldots ,\wp _n \in R\}\). If F, G and H are three generalized skew-derivations on R associated to the same automorphism \(\beta \) such that

$$ H\left( \xi \right) \xi -F(\xi )G(\xi )=0 $$

for each \(\xi \in S\). Let \(g_1,g_2,g_3 \) be the associated skew derivations respectively of H, F and G,  such that \(g_1,g_2,g_3\) are commuting with \(\beta \). Then we shall give the structure of H, F and G.