  We present non-critical Bianchi type $I$ string cosmology solutions in the
presence of central charge deficit term $\Lambda$. The leading order string
frame curvature appears to be in the high curvature limit $R\alpha'\gtrsim1$,
which underlines the necessity of including higher order $\alpha'$-corrections.
We give new solutions of two-loop (order $\alpha'$) $\beta$-function equations
of $\sigma$-model with non-zero $\Lambda$ and dilaton field in both cases of
absence and presence of spatially homogeneous $H$-field ($H=dB$). Also, the
evolution of solutions is studied in the Einstein frame, where the string
effective action can transform to Gauss-Bonnet gravity model coupled to the
dilaton field with potential. We study explicit examples in order $\alpha'$
with chosen values of appeared constants in the solutions and discuss the
cosmological implications.
