The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

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