The first part of the thesis is a joint work with Sukjoo Lee. It was shown by Diaconescu, Donagi and Pantev that Hitchin systems of type ADE are isomorphic to certain Calabi-Yau integrable systems. In this paper, we prove an analogous result in the setting of meromorphic Hitchin systems of type A which are known to be Poisson integrable systems. We consider a symplectization of the meromorphic Hitchin integrable system, which is a semi-polarized integrable system in the sense ofKontsevich and Soibelman. On the Hitchin side, we show that the moduli space of unordered diagonally framed Higgs bundles forms an integrable system in this sense and recovers the meromorphic Hitchin system as the fiberwise compact quotient. Then we construct a new family of quasi-projective Calabi-Yau threefolds and show that its relative intermediate Jacobian fibration, as a semi-polarized integrable system, is isomorphic to the moduli space of unordered diagonally framed Higgs bundles. The second part of the thesis studies the relation between the moduli spaces of modules over the sheaf of even Clifford algebra and the Prym variety, both associated to a conic bundle. In particular, we construct a rational map from the moduli space of modules over the sheaf of even Clifford algebra to the special subvarieties in Prym varieties, and check that the rational map is birational in some cases. As an application, we get an explicit correspondence between instanton bundles on cubic threefolds and twisted Higgs bundles.