= Solution
For the specified time operator, $N_kE_k=3ikqE_k$, because the derivatives of $e^{i(kx+k^3t)}$ cancel between $\partial_t$ and $\partial_x^3$. Expanding the derivatives of $E_kA_k$ gives
$$
N_k(E_kA_k)=E_k\left[\int_L\frac{N_l\varphi(l)}{k+l}\,d\lambda(l)
+3ik\int_L\frac{M_l\varphi(l)}{k+l}\,d\lambda(l)-3kI_x\right].
$$
For completeness, the extra derivative terms before simplification are $3ikA_{k,xx}-3k^2A_{k,x}+3ikqA_k$. Substituting $\varphi_{xx}+q\varphi=M_l\varphi+il\varphi_x$ turns their derivative contribution into $-3k\int_L\varphi_x\,d\lambda=-3kI_x$.
Applying $N_k$ to the original <linear integral equation> and using $I_x=-q$ cancels its right side. Thus
$$
\mathcal F(N\varphi)=3kE_k\int_L\frac{M_l\varphi(l)}{k+l}\,d\lambda(l).
$$
Part (a) gives $E_k\int_LM_l\varphi/(k+l)\,d\lambda=iM_k\varphi$. Therefore the exact requested identity is
$$
\boxed{\mathcal F(N\varphi)=3ikM\varphi.}
$$
Since $M\varphi=0$, this becomes homogeneous too. The same uniqueness argument gives $N\varphi=0$; it is not necessary to assume a separate uniqueness theorem for the differential time equation.
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