X[z_] := {U[z], V[z], Subscript[T, zz][z], Subscript[T, rz][z]};
A[z_] := {{0, -λ*ξ*σ^-1, σ^-1, 0}, {ξ, 0, 0, μ^-1}, {0, 0, 0, -ξ}, {0, 4 μ*η*ξ^2*σ^-1, λ*ξ*σ^-1, 0}};
system = X'[z] == A[z].X[z];
DSolve[system /. {σ -> λ + 2 μ, η -> λ + μ}, {U, V, Subscript[T, zz], Subscript[T, rz]}, z] // FullSimplify

sol = First@ DSolve[system /. {σ -> λ +  2 μ, η -> λ + μ},
    {U, V, Subscript[T, zz], Subscript[T, rz]}, z];

basis = sol /. (Thread[Array[C, 4] -> #] & /@ IdentityMatrix[4]);
Through[{U, V, Subscript[T, zz], Subscript[T, rz]}[z]] /. basis // Simplify
(* long output omitted *)

Eigensystem[A[z]/.{[Sigma] -> [Lambda] +
   2 [Mu], [Eta] -> [Lambda] + [Mu]}]//FullSimplify//MatrixForm