New infinite-dimensional multiple symmetry groups for the EMDA theory with two vector fields

The symmetry structures of stationary axisymmetric Einstein-Maxwell-Dilaton-Axion theory with 2 vector fields (EMDA-2 theory, for brevity) are further studied. By using the so-called extended double (ED)-complex function method, the usual Riemann-Hilbert (RH) problem is extended to an ED-complex formulation. Two pairs of ED RH transformations are constructed and they are verified to give infinite-dimensional multiple symmetry groups of the EMDA-2 theory, each of these symmetry groups has the structure of semidirect product of Kac-Moody group and Virasoro group. Moreover, the infinitesimal forms of the given ED RH transformations are calculated out, and these infinitesimal RH transformations give exactly the same infinitesimal symmetry transformations of the EMDA-2 theory as constructed in one of our previous papers. These results demonstrate that the two pairs of ED RH transformations in this paper provide us with exponentiations of all the infinitesimal symmetries constructed in our previous paper. The finite forms of symmetry transformations given in the present paper are more important and useful for theoretic studies and new solution generation, etc.