  We introduce a measure of quantum correlations in the $N$-qubit quantum
system which is invariant with respect to the $SU(2^N)$ group of
transformations of this system. This measure is a modification of the quantum
discord introduced earlier and is referred to as the unitary or
$SU(2^N)$-invariant discord. Since the evolution of a quantum system is
equivalent to the proper unitary transformation, the introduced measure is an
integral of motion and is completely defined by eigenvalues of the density
matrix. As far as the calculation of the unitary invariant discord is rather
complicated computational problem, we propose its modification which may be
found in a simpler way. The case N=2 is considered in details. In particular,
it is shown that the modified SU(4)-invariant discord reaches the maximum value
for a pure state.
  {A geometric measure of the unitary invariant discord of an $N$-qubit state
is introduced and a simple formula for this measure is derived, which allows
one to consider this measure as a witness of quantum correlations.} The
relation of the unitary invariant discord with the quantum state transfer along
the spin chain is considered. We also compare the modified SU(4)-invariant
discord with the geometric measure of SU(4)-invariant discord of the two-qubit
systems in the thermal equilibrium states governed by the different
Hamiltonians.
