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Take the spectral measure to have $n$ point masses at $k_j=i{\kappa }_j$, with distinct ${\kappa }_j>0$ and positive weights $c_j$. SetEvaluating the linear integral equation at these points gives the explicit algebraic system and reconstructionThe 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 $A_{ij}=c_j{E}_i/({\kappa }_i+{\kappa }_j)$, $B=I+A$, $\tau =\det B$ and $\Lambda =\mathrm{diag}({\kappa }_1,\dots ,{\kappa }_n)$. Then $A_x=-\Lambda A$ and $\Lambda A+A\Lambda =E{c}^T$. Using the derivative of a determinant and cyclicity of the trace,Hence the finite-rank KdV dressing determinant givesPositivity 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 $A=\mathrm{diag}(E)C\mathrm{diag}(c)$ is diagonally similar to $\mathrm{diag}(\sqrt{cE})C\mathrm{diag}(\sqrt{cE})$. 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.