  Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and
the related Lie-algebraic structures, we study multi-dimensional dispersionless
integrable systems that describe conformal structure generating equations of
mathematical physics. An interesting modification of the devised Lie-algebraic
approach subject to spatial-dimensional invariance and meromorphicity of the
related differential-geometric structures is described and applied in proving
complete integrability of some conformal structure generating equations. As
examples, we analyze the Einstein-Weyl metric equation, the modified
Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first
and second conformal structure generating equations and the inverse first
Shabat reduction heavenly equation. We also analyze the modified Pleba\'nski
heavenly equations, the Husain heavenly equation and the general Monge equation
along with their multi-dimensional generalizations. In addition, we construct
superconformal analogs of the Whitham heavenly equation.
