Structure of the solutions of the matricial ODE: $\dot U(t) = -U^2 - BU$.

Question

Let $A$ and $B$ be a fixed $n \times n$ positive semi-definite matrices. Given an $n \times n$ matrix $U$, define another $n \times n$ matrix by $F(U) := U^2 + U B$. Consider the ODE

$$ \dot U(t) = -F(U(t)),\, U(0) = A. $$ Define $U_\infty := \lim_{t \to \infty} U(t)$.

Question. In terms of $B$, can anything be said about the eigenvalue decomposition of $U_\infty$ ?