A two-dimensional additive problem with an increasing number of terms

In this paper there is established an asymptotic formula for the number of simultaneous representations of two numbers as sums of an increasing number of terms involving a power function, i.e., an asymptotic (as $n\to\infty$) formula is found for the number of solutions in integers $x_i$, $0\le x_i\le p$, of the following system of diophantine equations:
$$ \begin{cases} x_1+x_2+\dots+x_n=N_1,\\ x_1^2+x_2^2+\dots+x_n^2=N_2. \end{cases} $$
The analysis is carried out as in the proof of a local limit theorem of probability theory and involves estimates of Weyl sums.