How to calculate critical damping of a system with two springs and a damper (or two springs and two dampers)?
Re-writing the strain-stress equation
$$
0 = E_1Ɛ + \frac{η(E_1+E_2)}{E_2}\dot{Ɛ} - \frac{η}{E_2}\dot{σ} - σ
$$
for displacement/restoring force variables
$$
0 = E_1x + \frac{η(E_1+E_2)}{E_2}\dot{x} + m\frac{η}{E_2}\dddot{x} + m\ddot{x}
$$
and applying the Laplace transform, we arrive at the equation
$$
0 = E_1 + \frac{η(E_1+E_2)}{E_2}s + ms^2 + m\frac{η}{E_2}s^3 \tag1
$$
The formula for root $s = \frac{-E_1E_2}{η(E_1+E_2)} $ is only valid when $m=0$, and for root $s = ±\sqrt{-\frac{E_1}{m}}$ is only valid when $η=0$. In a general case of nonzero $m, η$, the equation (1) has three roots. The general formula for cubic equation roots is complicated. To write it down, the Wikipedia article introduces two first-level &quot;determinants&quot; $Δ_0 = b^2-3ac$ and $Δ_1 = 2b^3-9abc+27a^2d$, where $a, b, c, d$ are the cubic equation coefficients: $ax^3+bx^2+cx+d=0$, Then it defines an &quot;upper-level&quot; determinant $C$, built with the &quot;first-level&quot; determinants,
$$
C=\sqrt[3]{{Δ_1 ± \sqrt{Δ_1^2 - 4Δ_0^3}}\over{2}}
$$
the sign ± is selected to be plus or minus in order to guarantee that $C$ is never zero, and the three roots are
$$
x_k = -{1\over{3a}}\left(b + ξ^kC+{{Δ_0}\over{ξ^kC}}\right), \, k \in \{0,1,2\}, \, ξ={{-1+\sqrt{-3}}\over2} \tag2
$$
For a parallel-serial-connected spring-damper-mass system of your question, $a = m\frac{η}{E_2}, b = m, c = \frac{η(E_1+E_2)}{E_2}, d = E_1$.
The equation (1) reduces to a second-order ODE describing the simple damped harmonic oscillator:

for a trivial case of $η=0$, where there is no damping at all, and
the spring E2 does not participate in the motion equations;
the other non-dissipative case is $η\to{\infty}$, the springs act
in parallel with the total stiffness of $E_1+E_2$;
with $E_2\to\infty$, the bottom spring is &quot;turned off&quot;, the system
has one spring and one damper in parallel, this system you consider in the
beginning of your question;
with $E_1=0$, the upper spring is &quot;turned off&quot;, the system has one
spring and one damper in series, the system also has overdamped,
underdamped and critically damped solutions that can be found with the
formulas from your link.

For non-zero, finite values of $η, E_0, E_1, m$, iff the system has degenerated roots for certain combinations of $η, E_0, E_1, m$ values, then one of its linearly independent solutions has the form $t·exp(x_kt)$, the root $x_k$ has a negative real part. This solution may be called the critically damped solution. To find these combinations of $η, E_0, E_1, m$ values, one must search for degenerate solutions of Eq. 1 with multiple roots. The formulas are certainly not that simple as the formulas for a damped harmonic oscillator, but the ODE for the mechanical system of eq. 1, by virtue of an electrical-mechanical analogy, is identical to the ODE of a circuit with a Cauer topology of the generic three-pole unbalanced T-section LC low pass filter, the latter being often discussed on electronics forums.