  The small-$x$ contributions to the Bjorken sum rule within unified picture
$ln^2x+$LO DGLAP for different input parametrisations $g_1^{NS}(x,Q_0^2)$ are
presented. Theoretical predictions for $\int_{0}^{0.003} g_1^{NS}(x,Q^2=10) dx$
are compared with the SMC small-$x$ data. Rough estimation of the slope
$\lambda$, controlling the small-$x$ behaviour of $g_1^{NS}\sim x^{-\lambda}$
from the obtained results and SMC data is performed. The crucial role of the
running coupling $\alpha_s=\alpha_s(Q^2/z)$ at low-$x$ is taken into account.
