On the accuracy of the normal approximation. II

Estimates are presented for the asymptotically exact constants in the estimates of the accuracy of the normal approximation for the distributions of sums of independent identically distributed random variables with finite moments of order $2+\delta$, $0<\delta<1$. Refined practically applicable estimates of the accuracy of the normal approximation are constructed in which the right-hand side is a sum of two summands, the first summand being the Lyapunov fraction with the absolute constant close to the asymptotically exact one, whereas the second summand decreases faster than $n^{-\delta/2}$. Explicit estimates and special “expansions” are given for the second summand.