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：通过利用解决代数 $ top$- riccati 方程式和计算某种降缩子空间 $top$- palindromic mexicle peneros之间的联系,我们获得了关于这两个问题的理论和计算结果。理论上,我们引入了各种条件,以避免在 $top$- palicroducric 方程式铅笔中出现moulus-one egenvalue, 以及存在 $top$- Ricatical 方程式解决方案的条件。计算上,我们用新的定购程序改进了正本运算法,并引入了新的算法,根据铅笔的四面化或以所寻求的降缩压子空间的正方形投影机整体表示,计算出美元top$- palindromicrodumic 铅笔的降缩缩放子空间。