Integer decomposition for polyhedra defined by nearly totally unimodular matrices

We call a matrix $A$ nearly totally unimodular if it can be obtained from a totally unimodular matrix $\tilde{A}$ by adding to each row of $\tilde{A}$ an integer multiple of some fixed row $a^{\ssf T}$ of $\tilde{A}$. For an integer vector $b$ and a nearly totally unimodular matrix $A$, we denote by $P_{A,b}$ the integer hull of the set ${x\in{\Bbb R}^n\mid Ax\leq b}$. We show that $P_{A,b}$ has the integer decomposition property and that we can find a decomposition of a given integer vector $x\in kP_{A,b}$ in polynomial time.