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A kind of `reduced' Lagrangian (RL) formulation of the problem of
multi-dimensional supergravity solutions, for a class of them describing
distributed' marginal systems of multi-intersecting branes at arbitrary
angles, is introduced. It turns out that all the classical information
regarding every such configuration is derivable from a corresponding
first order quadratic RL, which `on the solution' identically vanishes.
The solution for every such configuration with $N$ arbitrary distributions,
is found that, lies on an $N$ dimensional subspace (the $H$-surface) of
the target (or configuration) space, parametrised by $N$ independent harmonic
functions of the overall transverse space. It is observed that every such
RL introduces a metric on the target space thereby identifying the
$H$-surface as a geodesic and null surface. The geodesic and the
nullity equations of this surface then follow directly from the field
equations for that RL. For orthogonal configurations, this approach
provides a simple derivation of the well known `superposition rule' of the
orthogonal solutions, together with the corresponding `intersection rules'.
For branes at angles, it leads to a new solution describing a
coniguration of $(p,p)$-branes at $SU(2)$ angles.
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