  We demonstrate that the dispersionless $\bar\partial$-dressing method
developed before for general heavenly equation is applicable to the $4+4$ and
$2N+2N$ - dimensional symmetric heavenly type equations. We introduce
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 $4+4$-dimensional equation to
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 $2N+2N$-dimensional case.
