http://tutorial.math.lamar.edu/Classes/DE/RepeatedEigenvalues.aspx
This gives an example of how to solve such systems. But I have a problem. what if the eigenspace due to an eigenvalue has dimension greater than one? Then there is more than one potential eigenvector (letter which looks like $n$). Will I have to get more solutions using that?
What the linked page neglects to mention is that it’s possible even for a $2\times2$ matrix to have an eigenvalue with both algebraic and geometric multiplicity of $2$. (If it’s a $2\times2$ matrix, then it must be a multiple of the identity.) In that case you can find a pair of linearly independent eigenvectors with that eigenvalue and just use the basic solution as described in an earlier section: for each eigenvector $\vec\eta$, you have a term of the form $ce^{\lambda t}\vec\eta$.