The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation KM + MK = r s^T we introduce a scalar function S^{(i, j)} = s^T K^j (I + M)⁻¹Kⁱr which is defined as same as in discrete case. S^{(i, j)} satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on r and s, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of S^{(i, j)} defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property S^{(i, j)} = S^(j,i). The solution M provides t function by t = ∣I + M∣. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.

We establish an infinite family of solutions in terms of elliptic functions of the lattice Boussinesq systems by setting up a direct linearisation scheme, which provides the solution structure for those equations in the elliptic case. The latter, which contains as main structural element a Cauchy kernel on the torus, is obtained from a dimensional reduction of the elliptic direct linearisation scheme of the lattice Kadomtsev–Petviashvili equation, which requires the introduction of a novel technical concept, namely the ‘elliptic cube root of unity’. Thus, in order to implement the reduction we define, more generally, the notion ofelliptic $$N{\textrm{th}}$$Nth root of unity, and discuss some of its properties in connection with a special class of elliptic addition formulae. As a particular concrete application we present the class of elliptic multi-soliton solutions of the lattice Boussinesq systems.