Real systems of ordinary differential equations \[ \frac{d\xi} {dt} =Q(\xi), \quad \xi, Q(\xi) \in \mathbb R^r, \] are investigated. \(Q(\xi)\) is a \(C^l\) function in some neighborhood of zero, \(l > 0\), \(Q(0) = 0\), and the matrix \(A = Q'(0)\) has no eigenvalues with zero real part. On a set of systems of ordinary differential equations described above, local smooth transformations preserving linear automorphism \(\xi=B\eta\) of these equations are considered. It is supposed, that the matrix \(B\) linear automorphism satisfies to an orthogonal condition, that is the spectrum of the matrix \(B\) lies on the unit circle and the Jordan form of \(B\) is a diagonal matrix. Sufficient conditions for local smooth equivalence, normalization and linearization of systems of ordinary differential equations with automorphism preserved are received.