Let $\mathbf{A},\mathbf{B},\mathbf{\mathbf{D}}$ be the regular matrices $(m,m)$, $\mathbf{C},\mathbf{E},\mathbf{X},\mathbf{Y}$ be the column vectors $(m,1)$, and $\alpha$ is the positive constant.
Is there any algebraic solution of the following system of two non-linear matrix equations
\begin{align*} (\mathbf{A}+\alpha\mathbf{B}\mathbf{Y}\mathbf{Y}^{T}\mathbf{B})\mathbf{X} & =\mathbf{C},\\ (\mathbf{D}+\alpha\mathbf{B}\mathbf{X}\mathbf{X}^{T}\mathbf{B})\mathbf{Y} & =\mathbf{E}, \end{align*}
for the unknown variables $\mathbf{X},\mathbf{Y}$?
I have tried a simple approach \begin{align*} \mathbf{X} & =(\mathbf{A}+\alpha\mathbf{B}\mathbf{Y}\mathbf{Y}^{T}\mathbf{B})^{-1}\mathbf{C}.\\ \end{align*} \ Then, \begin{align*} \mathbf{X}^{T}\mathbf{X} & =[(\mathbf{A}+\alpha\mathbf{B}\mathbf{Y}\mathbf{Y}^{T}\mathbf{B})^{-1}\mathbf{C}]^{T}[(\mathbf{A}+\alpha\mathbf{B}\mathbf{Y}\mathbf{Y}^{T}\mathbf{B})^{-1}\mathbf{C}],\\ \end{align*} with the following substitution \begin{align*} (\mathbf{D}+\alpha\mathbf{B}[(\mathbf{A}+\alpha\mathbf{B}\mathbf{Y}\mathbf{Y}^{T}\mathbf{B})^{-1}\mathbf{C}]^{T}[(\mathbf{A}+\alpha\mathbf{B}\mathbf{Y}\mathbf{Y}^{T}\mathbf{B})^{-1}\mathbf{C}]\mathbf{B})\mathbf{Y} & =\mathbf{E}.\\ \end{align*}
The equation might be simplified, but I am not sure how to continue, since the resulting equation for $\mathbf{X}, \mathbf{Y}$ will be non-linear.
Thank you very much for your help.