  We apply a nested variant of multigrade efficient congruencing to estimate
mean values related to that of Vinogradov. We show that when $\varphi_j\in
\mathbb Z[t]$ $(1\le j\le k)$ is a system of polynomials with non-vanishing
Wronskian, and $s\le k(k+1)/2$, then for all complex sequences $(\mathfrak
a_n)$, and for each $\epsilon>0$, one has \[ \int_{[0,1)^k} \left| \sum_{|n|\le
X} {\mathfrak a}_n e(\alpha_1\varphi_1(n)+\ldots +\alpha_k\varphi_k(n))
\right|^{2s} {\rm d}{\boldsymbol \alpha} \ll X^\epsilon \left( \sum_{|n|\le X}
|{\mathfrak a}_n|^2\right)^s. \] As a special case of this result, we confirm
the main conjecture in Vinogradov's mean value theorem for all exponents $k$,
recovering the recent conclusions of the author (for $k=3$) and Bourgain,
Demeter and Guth (for $k\ge 4$). In contrast with the $l^2$-decoupling method
of the latter authors, we make no use of multilinear Kakeya estimates, and thus
our methods are of sufficient flexibility to be applicable in algebraic number
fields, and in function fields. We outline such extensions.
