Linear System O.D.E.: 1<dim(Eigenspace)<eigenvalue multiplicity

Question

Let’s be:

• the linear system $\boldsymbol{x}’=A\boldsymbol{x}$ where $A\in M_{n\times n}(\mathbb{R})$ is constant and
• an eigenvalue ($\lambda$) with multiplicity $m>1$.

What happens if the eigenspace of $A$ corresponding to the eigenvalue $\lambda$ has dimension $k$, where $1<k<m$? If we have $k$ ($1<k<m$) linearly independent eigenvectors ($\boldsymbol{u_1},\dots,\boldsymbol{u_k}$), how can I construct $m$ inependent solutions of the O.D.E. $\boldsymbol{x}’=A\boldsymbol{x}$?

I know what to do:

• if $k=1$ (make a chain with generalized eigenvectors starting from the $\boldsymbol{u_1}$ eigenvector e.t.c.),
• if $k=m$ (make $m$ independent solutions $\boldsymbol{u_1}e^{\lambda t},\dots,\boldsymbol{u_m}e^{\lambda t}$.

I don't know what to do if $1<k<m$.

Thanks in advance!