System of inequalities

Let $N_{k,n,r}(P)$ be a number of integer solutions of the system of inequalities
$$ |x_1^\nu+\dots+x_k^\nu-y_1^\nu-\dots-y_k^\nu|\le P^{\nu-r},\ \ 1\le\nu\le n;\quad1\le x_1,\dots,x_k,y_1,\dots,y_k\le P. $$
The main result is the following estimate for $k-\frac{n^2}4\gg nr\log r$
$$ N_{k,n,r}(P)\ll P^{2k-\frac{n(n+1)}2+\frac{(n-r)(n-r+1)}2}. $$
This estimate has the right order with respect to $P$. For $r=n$ this is the classical Vinogradov mean value theorem.