So the method that I know of tackling second order 2x2 systems of differential equations is to reduce it to a first 4x4 system. But once you do so and solve for the eigenvalues and eigenvectors, how does one recover the 2x2 fundamental matrix? Especially when there are only 2 eigenvalues with a multiplicity of 2 and thus 2 4-component eigenvectors?
Here is an example of such an equation:
$$\mathbf{x''} = \begin{bmatrix} 2 & -5\\ 2 & -2 \end{bmatrix}\mathbf{x}$$
Not an answer just a comment but I think an elaboration of this idea leads to an answer :
Suppose that you have the second order system of differential equations $$\mathbf{x}''=A\mathbf{x}+B\mathbf{x}',$$then the substitution $\mathbf{y}=\mathbf{x}'$ transforms the system to a first-order system of differential equations, $$\mathbf{y}'=\mathbf{x}''=A\mathbf{x}+B\mathbf{y},$$ Then rewriting in terms the block-matrix, we have $$\begin{bmatrix}\mathbf{x}'\\\mathbf{y}'\end{bmatrix}=\begin{bmatrix}O&I\\A&B\end{bmatrix}\begin{bmatrix}\mathbf{x}\\\mathbf{y}\end{bmatrix}$$
Since $$\mathbf{x}''=\begin{bmatrix}2&-5\\2&-2\end{bmatrix}\mathbf{x}=A\mathbf{x},$$ Then, $$\begin{bmatrix}\mathbf{x}'\\\mathbf{y}'\end{bmatrix}=\begin{bmatrix}0&0&|&1&0\\0&0&|&0&1\\-&-&|&-&-\\2&-5&|&0&0\\2&-2&|&0&0\end{bmatrix}\begin{bmatrix}\mathbf{x}\\\mathbf{y}\end{bmatrix}$$ It is a linear system of first-order of differential equations.