By exploiting the connection between solving algebraic $\top$-Riccati equations and computing certain deflating subspaces of $\top$-palindromic matrix pencils, we obtain theoretical and computational results on both problems. Theoretically, we introduce conditions to avoid the presence of modulus-one eigenvalues in a $\top$-palindromic matrix pencil and conditions for the existence of solutions of a $\top$-Riccati equation. Computationally, we improve the palindromic QZ algorithm with a new ordering procedure and introduce new algorithms for computing a deflating subspace of the $\top$-palindromic pencil, based on quadraticizations of the pencil or on an integral representation of the orthogonal projector on the sought deflating subspace.

翻译：通过利用代数 $\top$-Riccati 方程求解与 $\top$-palindromic 矩阵束特定收缩子空间计算之间的联系，我们在两个问题上均获得了理论与计算成果。理论上，我们引入了避免 $\top$-palindromic 矩阵束出现模长为一的特征值的条件，以及 $\top$-Riccati 方程解存在的条件。计算上，我们通过一种新的排序过程改进了 palindromic QZ 算法，并基于矩阵束的二次化或所求收缩子空间上正交投影的积分表示，提出了计算 $\top$-palindromic 矩阵束收缩子空间的新算法。