Let
$I_{s,k,r}(X)$
denote the number of integral solutions of the modified Vinogradov system of equations
$$\begin{eqnarray}x_{1}^{j}+\cdots +x_{s}^{j}=y_{1}^{j}+\cdots +y_{s}^{j}\quad (1\leqslant j\leqslant k,\;j\neq r),\end{eqnarray}$$
with
$1\leqslant x_{i},y_{i}\leqslant X\;(1\leqslant i\leqslant s)$
. By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for
$I_{s,k,r}(X)$
for
$1\leqslant r\leqslant k-1$
. In particular, when
$s,k\in \mathbb{N}$
satisfy
$k\geqslant 3$
and
$1\leqslant s\leqslant (k^{2}-1)/2$
, we establish the essentially diagonal behaviour
$I_{s,k,1}(X)\ll X^{s+\unicode[STIX]{x1D700}}$
.