= Solution
Take the spectral measure to have $n$ point masses at $k_j=i\kappa_j$, with distinct $\kappa_j>0$ and positive weights $c_j$. Set
$$
E_j=e^{-\kappa_jx+\kappa_j^3t},\qquad \Phi_j=\varphi(i\kappa_j,x,t).
$$
Evaluating the <linear integral equation> at these points gives the explicit algebraic system and reconstruction
$$
\boxed{\Phi_i+\sum_{j=1}^n\frac{c_jE_i}{\kappa_i+\kappa_j}\Phi_j=E_i,\qquad
q=-\partial_x\sum_{j=1}^nc_j\Phi_j.}
$$
The sign is fixed by $i/(i\kappa_i+i\kappa_j)=1/(\kappa_i+\kappa_j)$. No continuum integration remains.
There is also a useful determinant form. Define $A_{ij}=c_jE_i/(\kappa_i+\kappa_j)$, $B=I+A$, $\tau=\det B$ and $\Lambda=\operatorname{diag}(\kappa_1,\ldots,\kappa_n)$. Then $A_x=-\Lambda A$ and $\Lambda A+A\Lambda=Ec^T$. Using the derivative of a <determinant> and cyclicity of the <trace>,
$$
\partial_x\log\tau=-\operatorname{tr}(B^{-1}\Lambda A)
=-\tfrac12\operatorname{tr}[B^{-1}(\Lambda A+A\Lambda)]
=-\tfrac12c^TB^{-1}E.
$$
Hence the <finite-rank KdV dressing determinant> gives
$$
\boxed{q=2\partial_x^2\log\tau.}
$$
Positivity ensures a regular real solution: the <Cauchy matrix> $C_{ij}=1/(\kappa_i+\kappa_j)$ is the Gram matrix of the independent functions $e^{-\kappa_i s}$ on $s>0$, so it is positive definite. The matrix $A=\operatorname{diag}(E)C\operatorname{diag}(c)$ is diagonally similar to $\operatorname{diag}(\sqrt{cE})C\operatorname{diag}(\sqrt{cE})$. Its eigenvalues are positive, and $\tau>0$. The resulting reflectionless <multisoliton solution> has $n$ distinct scales and velocities; the weights fix their positions. Arbitrary signed or complex weights need not give a nonsingular real solution.
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