Invariant coordinate subspaces of normal form of a system of ordinary differential equations

We consider a system of ordinary differential equations (ODEs) with nondegenerate linear part near its stationary point in two cases: in general case and in Hamiltonian case. For these two cases the problem of existence of an invariant coordinate subspace in the coordinates of normal form is considered. The theorems of existence of invariant coordinate subspaces with explicit conditions are proven. Some examples with different cases of resonances between eigenvalues of the linear part of the system of ODE are considered. The technique for determination of resonance relations with the help of $q$-subdiscriminants is presented. An example of determination of resonance relations is given for a certain model system with six degrees of freedom.