  This paper concerns some inverse eigenvalue problems of the quadratic
$\star$-(anti)-palindromic system $Q(\lambda)=\lambda^2 A_1^{\star}+\lambda A_0
+ \epsilon A_1$, where $\epsilon=\pm 1$, $A_1, A_0 \in \mathbb{C}^{n\times n}$,
$A_0^{\star}=\epsilon A_0$, $A_1$ is nonsingular, and the symbol $\star$ is
used as an abbreviation for transpose for real matrices and either transpose or
conjugate transpose for complex matrices. By using the spectral decomposition
of the quadratic $\star$-(anti)-palindromic system, the inverse eigenvalue
problems with entire/partial eigenpairs given, and the model updating problems
with no-spillover are considered. Some conditions on the solvabilities of these
problems are given, and algorithms are proposed to find these solutions. These
algorithms are illustrated by some numerical examples.
