On integer programing with restricted determinants

Let $A$ be an $(m \times n)$ integral matrix of the rank $n$ and let $P=\{ x : A x \leq b\}$ be $n$-dimensional polytope. Width of $P$ is defined as $ w(P)=min\{max_P x^Tu - min_P x^Tv:\ x\in \mathbb{Z}^n\setminus\{0\} \}$. Let $\Delta$ denote the smallest absolute value of the determinant among basis matrices of $A$. We prove that if every basis matrix of $A$ has determinant equal to $\pm \Delta$ and $w(P)\ge (\Delta-1)(n+1)$, then $P$ contains a lattice $n$ - dimensional simplex. When $P$ is a simplex and $w(P)\ge \Delta-1$ we describe the polynomial time algorithm for finding an integer point in $P$.