On the exponents of the convergence of singular integrals and singular series of a multivariate problem

In the paper we continue studies on the theory of multivariate trigonometric sums, in the base of which lies of the I.M.Vinogradov's method. Here we obtain for $n=r=2$ lower estimates of the convergence exponent of the singular series and the singular integral of the asymptotic formulas for $P\to\infty$ for the number of solutions of the following system of Diophantine equations
$$\sum_{j=1}^{2k}(-1)^jx_{1,j}^{t_1}\dots x_{r,j}^{t_r}=0, 0\leq t_1,\dots, t_r\leq n,$$
where $n\geq 2,r\geq 1, k$ are natural numbers, moreover an each variable $x_{i,j}$ can take all integer values from $1$ to $P\geq 1.$