  In the present and accompanying papers we explicitly construct the on-shell
supersymmetric component actions for 3-branes moving in D=6 and in D=8 within
the nonlinear realizations framework.
  In the first paper we apply our schema to construct the action of
supersymmetric 3-brane in D=6. It turns out that all ingredients entering the
component action can be obtained almost algorithmically by using the nonlinear
realizations approach. Within this approach, properly adapted to the
construction of on-shell component actions, we pay much attention to broken
supersymmetry. Doing so, we were able to write the action in terms of purely
geometric objects (vielbeins and covariant derivatives of the physical bosonic
components), covariant with respect to broken supersymmetry. It turns out that
all terms of the higher orders in the fermions, are hidden inside these
covariant derivatives and vielbeins. Moreover, the main part of the component
action just mimics its bosonic cousin in which the ordinary space-time
derivatives and the bosonic world volume are replaced by their covariant
supersymmetric analogs. The Wess-Zumino term in the action, which does not
exist in the bosonic case, can be also easily constructed in terms of reduced
Cartan forms. Keeping the broken supersymmetry almost explicit, one may write
the Ansatz for the component action, fully defined up to two constant
parameters. The role of the unbroken supersymmetry is just to fix these
parameters.
