The limit transition $\mathbb{P}_2\to\mathbb{P}_1$

The way which allow to consider the well known limit transition $\mathbb{P}_2\to\mathbb{P}_1$ as a double asymptotic of solutions of equation $\mathbb{P}_2$ in a special “transition” domain which is characterized by the relation $\alpha^2/x^3$, where $\alpha$ is the coefficient of $\mathbb{P}_2$, and $x$ is its argument is found. The importance of Bäcklund transformation for this limit transition is clarified. This limit is studied for all possible solutions of $\mathbb{P}_2$.