  Two-dimensional (2D) semiconductor structures of materials without inversion
center (e.g. zinc-blende ${\rm A^{III}B^V}$) possess the zero-field conduction
band spin-splitting (Dresselhaus term), which is linear and cubic in wavevector
$k$, that arises from cubic in $k$ splitting in a bulk material. At low carrier
concentration the cubic term is usually negligible. However, if we will be
interested in the following dimensional quantization (in 2D plane) and the
character width in this direction is comparable with the width of 2D-structure,
then we have to take into account $k^3$-terms as well (even at low
concentrations), that after quantization leads to comparable contribution that
arises from $k$-linear term. We propose the general procedure for derivation of
Dresselhaus spin-splitting Hamiltonian applicable for any curvilinear
1D-structures. The simple examples for the cases of a quantum wire (QWr) and a
quantum ring (QR) defined in usual [001]-grown 2D-structure are presented.
