= Solution
Set $\omega_1=\omega_2=\omega$. Define the <quantum rotating frame> by
$$
H_Z=\frac{\omega}{2}(Z\otimes I+I\otimes Z),\qquad
R(t)=e^{-itH_Z},\qquad
|\psi_{\mathrm r}(t)\rangle=R(t)^\dagger|\psi(t)\rangle.
$$
The isotropic <Heisenberg model> coupling commutes with the total $z$ <spin angular momentum>, so $[H_Z,C]=0$. The transformed <Hamiltonian operator> is therefore
$$
H_{\mathrm r}=R^\dagger H_{\mathrm H}R-iR^\dagger\dot R
=R^\dagger H_{\mathrm H}R-H_Z=\frac J2C.
$$
Here removal of the common <Zeeman energy> is exact; the exchange is resonant and cannot be discarded using the <weak coupling condition for two NMR spins>.
Put $a=Jt$, $c=\cos a$, and $s=\sin a$. Exponentiating the parallel-spin entries and the middle $2\times2$ <matrix> gives the <time-evolution operator>
$$
\boxed{U(t)=e^{-itH_{\mathrm r}}=
\begin{pmatrix}
e^{-ia/2}&0&0&0\\
0&e^{ia/2}c&-ie^{ia/2}s&0\\
0&-ie^{ia/2}s&e^{ia/2}c&0\\
0&0&0&e^{-ia/2}
\end{pmatrix}.}
$$
For example, at $a=\pi/2$ this equals the <SWAP gate> times $e^{-i\pi/4}$, which checks both the exchange angle and the overall phase.
The printed network reads from left to right as $U$, a <Pauli Z gate> on the \b[upper spin only], $U$, and another upper-spin <Pauli Z gate>. Let
$$
P=Z\otimes I=\operatorname{diag}(1,1,-1,-1).
$$
Its total <time-evolution operator>, with the first operation on the right, is $W=PUPU$. For this <Heisenberg exchange echo>,
$$
PH_{\mathrm r}P=\frac J2(-X\otimes X-Y\otimes Y+Z\otimes Z).
$$
The three paired <Pauli operators> $X\otimes X$, $Y\otimes Y$, $Z\otimes Z$ commute pairwise: for example both products $(X\otimes X)(Y\otimes Y)$ and $(Y\otimes Y)(X\otimes X)$ equal $-Z\otimes Z$. Thus $PH_{\mathrm r}P$ commutes with $H_{\mathrm r}$, and
$$
\boxed{W=e^{-itPH_{\mathrm r}P}e^{-itH_{\mathrm r}}
=e^{-iJtZ\otimes Z}
=\begin{pmatrix}
e^{-ia}&0&0&0\\
0&e^{ia}&0&0\\
0&0&e^{ia}&0\\
0&0&0&e^{-ia}
\end{pmatrix}.}
$$
The <spin echo> cancels the $XX+YY$ exchange terms and retains the <Ising coupling>. Equivalently, over the total free-evolution time $2t$ it produces the effective <Hamiltonian operator> $JZ\otimes Z/2$ exactly for ideal instantaneous gates. It is generally \b[not the identity]. A physical $\pi$ rotation about $z$ is $e^{-i\pi Z/2}=-iZ$; using two such pulses instead of the ideal $Z$ gates changes $W$ only by a physically irrelevant <global phase> of minus one.
The implementation requires <frequency-selective spin control>. Under the usual common-<radio frequency> <NMR> assumptions, equal resonance frequencies give no way to apply a pulse to the first spin while leaving the second untouched. For equally coupled spins in a spatially uniform control field, the transverse control <Hamiltonian operator> has the form
$$
H_c(t)=\frac{u_x(t)}2(X\otimes I+I\otimes X)
+\frac{u_y(t)}2(Y\otimes I+I\otimes Y).
$$
Both this control and the drift <Hamiltonian operator> commute with the <SWAP gate> $S$. Every reachable <time-evolution operator> consequently commutes with $S$. But $SPS=I\otimes Z\ne P$: the individual pulse required by the diagram fails this symmetry test. This is the <exchange symmetry obstruction to individual spin control>. Applying simultaneous $Z$ gates gives $Q=Z\otimes Z$, which commutes with $H_{\mathrm r}$ and instead yields $QUQU=U^2$; it does not refocus the exchange. A common change of the RF phase reference also supplies only a collective rotation.
This obstruction assumes the usual common-RF control with no extra handle distinguishing the spins. Equal resonance frequencies do not universally forbid individual addressing: a spatially selective field, unequal control couplings, or another symmetry-breaking interaction could alter <controllability>. Such resources are absent from the specified equivalent-spin <NMR> model.
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