NOVELTY - Operating a reactor comprising a first reactor, comprises (i) electrically feeding the first reactor, feeding and distributing a mixture of water vapor and carbon dioxide to each cathode, and feeding and distributing carbon dioxide to the cathode of the other of the two elementary cells, performing a high temperature electrolysis of water vapor at each cathode, or performing high temperature co-electrolysis of water vapor and carbon dioxide, and (ii) feeding and distributing a mixture of hydrogen and carbon monoxide to each cathode and performing methanation by heterogeneous catalysis.    USE - The methods are useful for operating a reactor and producing methane (all claimed).    ADVANTAGE - The method is energy saving and economical.    DETAILED DESCRIPTION - Operating a reactor comprising a first reactor, comprises (i) electrically feeding the first reactor, feeding and distributing water vapor or a mixture of water vapor and carbon dioxide to each cathode, or feeding and distributing water vapor to the cathode of one of two adjacent elementary cells, and feeding and distributing carbon dioxide to the cathode of the other of the two elementary cells, performing a high temperature electrolysis of water vapor at each cathode, or performing high temperature co-electrolysis of water vapor and carbon dioxide, and (ii) feeding and distributing a mixture of hydrogen and carbon monoxide, or a mixture of hydrogen and carbon dioxide to each cathode and performing methanation by heterogeneous catalysis at each cathode. The first reactor comprises a stack of solid oxide electrolyzer cell (SOEC) type elementary electrolysis cells, each formed of a cathode, an anode and an electrolyte, the electrolyte is interposed between the cathode and the anode, and multiple interconnectors each arranged between electric and fluid medium of two unit cells, adjacent with one of its faces in electrical contact with the anode to one of two unit cells and the other side in electrical contact with the cathode of the other two cells of elementary cathodes in material catalyst for methanation reaction. An INDEPENDENT CLAIM is also included for producing methane by implementing the operating method, comprising supplying methane product at the outlet of the first reactor, performing methanation by adapting a second reactor, when the methane conversion rate at the outlet of the first reactor is less than a threshold value, the methane product is fed to the output of the second reactor, or directly feeding the methane product into at the outlet of the first reactor, a storage tank or a distribution system.