On real solutions of systems of equations

Systems of equations $f_1=\cdots=f_{n-1}=0$ в $\mathbb R^n=\{x\}$ in $\mathbb R^n=\{x\}$ having the solution $x=0$ are considered under the assumption that the quasi-homogeneous truncations of the smooth functions $f_1=\cdots=f_{n-1}$ are independent at $x\ne0$. It is shown that, for $n\ne2$ and $n\ne4$, such a system has a smooth solution which passes through $x=0$ and has nonzero Maclaurin series.