Extremal polynomials connected with Zolotarev polynomials

Let two points $a$ and $b$ be given on the real axis, located to the right and left of the segment $[-1, 1]$ respectively. The extremal problem is posed: find an algebraic polynomial of n-th degree, which at the point a takes value $A$, on the segment $[-1, 1]$ does not exceed $M$ in modulus and takes the largest possible value at $b$. This problem is related to the second problem of Zolotarev. In the article the set of values of the parameter $A$ for which this problem has a unique solution is indicated, and an alternance characteristic of this solution is given. The behavior of the solution with respect to the parameter $A$ is studied. It turns out that for some $A$ the solution can be obtained with the help of the Chebyshev polynomial, while for all other admissible $A$ - with the help of the Zolotarev polynomial.