  Admissible structure constants related to the dual Lie superalgebras of
particular Lie superalgebra $({\cal C}^3 + {\cal A})$ are found by
straightforward calculations from the matrix form of super Jacobi and mixed
super Jacobi identities which are obtained from adjoint representation. Then,
by making use of the automorphism supergroup of the Lie superalgebra $({\cal
C}^3 + {\cal A})$, the Lie superbialgebra structures on the Lie superalgebra
$({\cal C}^3 + {\cal A})$ are obtained and classified into inequivalent 31
families. We also determine all corresponding coboundary and bi-r-matrix Lie
superbialgebras. The quantum deformations associated with some Lie
superbialgebras $({\cal C}^3 + {\cal A})$ are obtained, together with the
corresponding deformed Casimir elements. As an application of these quantum
deformations, we construct a deformed integrable Hamiltonian system from the
representation of the Hopf superalgebra ${{U}_{_\lambda}}^{\hspace{-1mm}({\cal
C}_{p=1}^{2,\epsilon} \oplus {\cal A}_{1,1})}\big(({\cal C}^3+{\cal A})\big)$.
