Paucity problems and some relatives of Vinogradov's mean value theorem

We consider relatives of the Vinogradov system of equations in which one or more equations have been deleted. In particular, when $k\geqslant 4$ and $0\leqslant d\leqslant (k-2)/4$, we consider the system of Diophantine equations
$$ x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d). $$
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 $d=o(k^{1/4})$. Analogous systems with more than one deleted equation will be discussed should time permit.