  We study solutions of the KdV equation governed by a stationary equation for
symmetries from the non-commutative subalgebra, namely, for a linear
combination of the master-symmetry and the scaling symmetry. The constraint
under study is equivalent to a sixth order nonautonomous ODE possessing two
first integrals. Its generic solutions have a singularity on the line $t=0$.
The regularity condition selects a 3-parameter family of solutions which
describe oscillations near $u=1$ and satisfy, for $t=0$, an equation equivalent
to degenerate $P_5$ equation. Numerical experiments show that in this family
one can distinguish a two-parameter subfamily of separatrix step-like solutions
with power-law approach to different constants for $x\to\pm\infty$. This gives
an example of exact solution for the Gurevich--Pitaevskii problem on decay of
the initial discontinuity.
