Integrability properties of a symmetric 4  +  4-dimensional heavenly-type equation

We demonstrate that the dispersionless -dressing method developed before for a general heavenly equation is applicable to the 4  +  4 and 2N  +  2N-dimensional symmetric heavenly-type equations. We introduce a generating relation and derive the two-form, defining the potential and equation for it. We develop the dressing scheme, calculate a class of special solutions and demonstrate that reduction from a 4  +  4-dimensional equation to a four-dimensional general heavenly equation can be effectively performed on the level of the dressing data. We consider also the extension of the proposed scheme to the 2N  +  2N-dimensional case. (paper)