A foliation is a global section of the differential sheaf tensorring  a  line bundle. It can also be regarded as a differential equation.  For example, a  fibration  on a surface gives a  foliation with a meromorphic first integral.   One can classify  all foliations by so-called Kodaira dimension.   In this talk, we will investigate the foliations with Kodaira dimension one (e.g., Riccati foliations). In this case, there is  an adjoint  fibration. We will  describe precisely all singular fibers of the  adjoint fibration.  As an application, we will prove that a Belyi  fibration with  two singular over $P^1$  gives a Riccati foliation and compute its Mordell-Weil group in a joint work with C. Gong and S.-L. Tan.   Furthermore, we can  classify such fibrations on a rational surface.

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