On formulation of modified problems for the Euler–Darboux equation with parameters equal to $\dfrac{1}{2}$ in absolute value

We consider the Euler–Darboux equation with parameters equal to $\dfrac{1}{2}$ in absolute value. Since the Cauchy problem in the classical formulation in ill-posed for such values of parameters, we propose formulations and solutions of modified Cauchy-type problems with the following values of parameters: a) $\alpha=\beta=\frac{1}{2},$ b) $\alpha=- \frac{1}{2},$ $\beta=- \frac{1}{2},$ c) $\alpha=\beta=- \frac{1}{2}.$ In the case а), the modified Cauchy problem is solved by the Riemann method. We use the obtained result to formulate the analog of the problem $\Delta_1$ in the first quadrant with shifted boundary-value conditions on axes and nonstandard conjunction conditions on the singularity line of the coefficients of the equation $y=x.$ The first condition is gluing normal derivatives of the solution and the second one contains limiting values of combination of the solution and its normal derivatives. The problem is reduced to a uniquely solvable system of integral equations.