[@Ch] firstly showed the strong well-posedness for SDE under the assumptions that $\sigma$ is Lipschitz continuous and $b$ is $\alpha$-Hölder continuous in $x$ and $\beta$-Hölder continuous in $\v$ with $\alpha\in (\frac{2}{3},1)$ and $\beta\in(0,1)$. The proofs in [@Ch] strongly depend on some explicit estimates for Kolmogorov operator with constant coefficients and Zvonkin’s transformation. In a recent joint work [@Wa-Zh] with F.Y. Wang, we also showed the strong uniqueness and homeomorphism property for under weaker Hölder-Dini’s continuity assumption on $b$. The proofs in [@Wa-Zh] rely on a characterization of Hölder-Dini’s spaces and gradient estimates for the semigroup associated with the kinetic operator. Notice that in [@Ch] and [@Wa-Zh], more general degenerate SDEs are considered, while, the case with critical differentiability indices $\alpha=\frac{2}{3}$ and $\beta=0$ is left open. The purpose of this work is to establish a similar theory for degenerate SDE as in Krylov and Röckner’s paper [@Kr-Ro] (see also [@Zh4]). In particular, the critical indices $\alpha=\frac{2}{3}$ and $\beta=0$ are covered. More precisely, we aim to prove that \[Main\] Suppose that for some $K\geq 1$ and all $(t,x,\v)\in\mR_+\times\mR^{2d}$, $$\begin{aligned}
K^{-1}|\xi|\leq |\sigma^*_t(x,\v)\xi|\leq K|\xi|,\ \  \forall\xi\in\mR^d,\tag{{\bf UE}}\end{aligned}$$ where $\sigma^*$ denotes the transpose of matrix $\sigma$, and for some $p>2(2d+1)$, $$\kappa_0:=\sup_{s\geq 0}\|\nabla\sigma_s\|^p_p+\int_0^\infty\|(\mI-\Delta_x)^{\frac{1}{3}} b_s\|_p^p\dif s<\infty.$$ Then 