Associated to oriented surfaces immersed in $\R^3,$ here are
studied pairs of transversal foliations
with singularities, defined on the {\it Elliptic} region, where the Gaussian curvature
$\mathcal K$, given by the product of the principal curvatures ${k_1},\; k_2$ of the immersion, is positive.
The {\it leaves} of the foliations are the {\it lines of} $
{\mathcal M}$-{\it mean curvature,}
along which the normal curvature of the immersion is given by a function
$ {\mathcal M}={\mathcal M}({k_1},{k_2})\in [{k_1}, k_2]$, called a $ {\mathcal M}$- {\it mean
curvature}, whose properties extend and unify those of the
{\it arithmetic} $ {\mathcal H}=({k_1}+k_2)/2$, the {\it geometric}
$\sqrt{\mathcal K}$ and {\it harmonic} ${\mathcal K}/{\mathcal H}=((1/{k_1} + 1/{k_2})/2)^{-1}$
{ \it classical mean
curvatures}.The {\it singularities} of the foliations are
the {\it umbilic points} and {\it parabolic curves}, where ${k_1} = k_2$ and ${\mathcal K} = 0$, respectively. Here are determined the patterns of $ {\mathcal M}$- {\it
mean curvature lines} near the
{\it umbilic points}, {\it parabolic curves} and
$ {\mathcal M}$-{\it mean curvature cycles} (the periodic
leaves of the foliations), which are structurally stable under small
perturbations of the immersion. The genericity of these patterns is also established.
These patterns provide the three essential local ingredients to
establish
sufficient
conditions, likely to be also necessary, for $ {\mathcal M}$-{\it Mean
Curvature Structural Stability} of immersed surfaces. This constitutes a natural unification
and complement for the results obtained previously by the
authors for the
{\it Arithmetic}, \cite {m}, {\it Asymptotic}, \cite {a1, a2},
{\it Geometric}, \cite {g} and {\it Harmonic}, \cite {h}, {\it classical} cases
of {\it Mean Curvature Structural Stability}.