[en] The present paper is the completion of experimental investigations of the π^-p → π⁰π⁰n reaction near the threshold aimed at verification of chiral theory of strong interactions. The experimental arrangement involving the 4-channel hodoscope of Cherenkov γ spectrometers and scintillator counters was used to measure the total near-threshold cross sections of the reaction studied. The total cross sections are (0.13+-0.02) and (0.032+-0.005) mb at Tsub(π)=240 and 200 MeV energies, respectively. Isotope analysis of total cross sections for the π^-p → π⁰π⁰n reactions at 270 and 240 MeV is made. It has shown that the total cross sections for all channels of the πN → ππN reaction are agreable in conformity with the requirements of the isotopic invariance of strong interactions. The ratio of the isotopic ampitudes X=F₁₀/F₃₂ equals 3.8+-0.6. Calculations show that experimental data on total cross sections of all channels of the πN → ππN reaction cannot be described by means of introducing into the effective Lagrangian of ππ scattering of additional terms corresponding to the fourth order of ππ scattering amplitude expansion in 4-momentum pions