  When $k\ge 4$ and $0\le d\le (k-2)/4$, we consider the system of Diophantine
equations \[ x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\le j\le k,\, j\ne
k-d). \] We show that in this cousin of a Vinogradov system, there is a paucity
of non-diagonal positive integral solutions. Our quantitative estimates are
particularly sharp when $d=o(k^{1/4})$.
