On the Vinogradov Additive Problem

Let us state the main result of the paper. Suppose that the collection $N_1,\dots,N_n$ is admissible. Then, in the representation
$$ \begin{cases} p_1+p_2+\dots+p_k=N_1, \\ \dots\dots\dots\dots\dots\dots\dots\dots \\ p_1^n+p_2^n+\dots+p_k^n=N_n, \end{cases} $$
where the unknowns $p_1,p_2,\dots,p_k$ take prime values under the condition $p_s>n+1$, $s=1,\dots,k$, the number $k$ is of the form
$$ k=k_0+b(n)s, $$
where $s$ is a nonnegative integer. Further, if $k_0\ge a$, then, in the representation for $k$, we can set $s=0$, but if $k_0\le a-1$, then, for a given $k_0$ there exist admissible collections $(N_1,\dots,N_n)$ that cannot be expressed as $k_0$ summands of the required form, but can be expressed as $k_0+b(n)$ summands.