  We introduce two basic invariant forms which define generic surface in
3-space uniquely up to Lie sphere equivalence. Two particularly interesting
classes of surfaces associated with these invariants are considered, namely,
the Lie-minimal surfaces and the diagonally-cyclidic surfaces. For
diagonally-cyclidic surfaces we derive the stationary modified Veselov-Novikov
equation, whose role in the theory of these surfaces is similar to that of
Calapso's equation in the theory of isothermic surfaces. Since Calapso's
equation itself turns out to be related to the stationary Davey-Stewartson
equation, these results shed some new light on differential geometry of the
stationary Davey-Stewartson hierarchy. Diagonally-cyclidic surfaces are the
natural Lie sphere analogs of the isothermally-asymptotic surfaces in
projective differential geometry for which we also derive the stationary
modified Veselov-Novikov equation with the different real reduction. Parallels
between invariants of surfaces in Lie sphere geometry and reciprocal invariants
of hydrodynamic type systems are drawn in the conclusion.
