q-connection of Levi-Civita

Question

Let $(M,g_q)$ be a q-riemannian manifold, $q\in Diff(M)$, $q.g(X_q,Y_q)=g(q.X_q,q.Y_q)$. We recall:

$$q.f(x)=f(qx)$$

$$q.X(f)=X(q.f)$$

We define the q-connection of Levi-Civita $\nabla$:

$$\nabla_{X_q}Y_q-\nabla_{Y_q}X_q=[X_q,Y_q]$$

$$X_q (g(Y_q,Z_q))=g(\nabla_{X_q}Y_q,q.Z_q)+g(q.Y_q,\nabla_{X_q}Z_q)$$

Does the q-connection of Levi-Civita exist and is unique?