Bifurcation of the Point of Equilibrium in Systems with Zero Roots of the Characteristic Equation

We consider the following real autonomous system of $2d$ differential equations with a small positive parameter $\varepsilon$:
$$\dot x_i=x_{i+d}+X_i^{(n+1)}(x,\varepsilon ),\qquad \dot x_{i+d}=-x_i^{2n-1}+X_{i+d}^{(2n)}(x,\varepsilon ),\qquad i=1,\dots,d,$$
where $d\ge 2$, $n\ge 2$, and the $X_j^{(k)}$ are continuous functions continuously differentiable with respect to $x$ and $\varepsilon$ the required number of times in the neighborhood of zero; their expansion begins with order $k$ if we assume that the variables $x_i$ are of first order of smallness, $\varepsilon$ is of second order, and the variables $x_{i+d}$ are of order $n$. We write out a finite number of explicit conditions on the coefficients of the lower terms in the expansion of the right-hand side of this system guaranteeing that for any sufficiently small $\varepsilon > 0$ the system has one or several $d$-dimensional invariant tori with infinitely small frequencies of motions on them.