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 P₅ 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 → ±∞. This gives an example of exact solution for the Gurevich–Pitaevskii problem on decay of the initial discontinuity.

对于非交换子代数的对称性，即对于主对称性和缩放对称性的线性组合，我们研究由平稳方程控制的KdV方程的解。研究的约束等效于具有两个第一积分的六阶非自治ODE。它的一般解在直线t = 0上具有奇异性。正则条件选择一个三参数组的解，它们描述u = 1附近的振动，并且对于t = 0，满足一个等价式P ₅的方程。方程。数值实验表明，在该族中，对于x →±∞，可以使用幂律方法将幂阶法区分为分离参数阶跃解的两参数子族。这给出了有关初始不连续性衰减的Gurevich-Pitaevskii问题的精确解的示例。