  Motivated by Ball, Li, Timotin and Trent's Schur-Agler class version of
commutant lifting theorem, we introduce a class, denoted by
$\mathcal{P}_n(\mathcal{H})$, of $n$-tuples of commuting contractions on a
Hilbert space $\mathcal{H}$. We always assume that $n \geq 3$. The importance
of this class of $n$-tuples stems from the fact that the von Neumann inequality
or the existence of isometric dilation does not hold in general for $n$-tuples,
$n \geq 3$, of commuting contractions on Hilbert spaces (even in the level of
finite dimensional Hilbert spaces). Under some rank-finiteness assumptions, we
prove that tuples in $\mathcal{P}_n(\mathcal{H})$ always admit explicit
isometric dilations and satisfy a refined von Neumann inequality in terms of
algebraic varieties in the closure of the unit polydisc in $\mathbb{C}^n$.
