  For the case of algebraic curves - compact Riemann surfaces - it is shown
that de Rham cohomology group $H^{1}_{\mathrm{dR}}(X,\mathbb{C})$ of a genus
$g$ Riemann surface $X$ has a natural structure of a symplectic vector space.
Every choice of a non-special effective divisor $D$ of degree $g$ on $X$
defines a symplectic basis of $H^{1}_{\mathrm{dR}}(X,\mathbb{C})$, consisting
of holomorphic differentials and differentials of the second kind with poles on
$D$. This result, the algebraic de Rham theorem, is used to describe the
tangent space to Picard and Jacobian varieties of $X$ in terms of differentials
of the second kind, and to define a natural vector fields on the Jacobian of
$X$ that move points of the divisor $D$. In terms of the Lax formalism on
algebraic curves, these vector fields correspond to the Dubrovin equations in
the theory of integrable systems, and the Baker-Akhierzer function is naturally
obtained by the integration along the integral curves.
