Non-homogeneous systems of hydrodynamic type possessing Lax representations

We consider 1+1 - dimensional non-homogeneous systems of hydrodynamic type that possess Lax representations with movable singularities. We present a construction, which provides a wide class of examples of such systems with arbitrary number of components. In the two-component case a classification is given.Comment: 22 pages, latex, minor change

An abstract approach to Loewner chains

We present a new geometric construction of Loewner chains in one and several complex variables which holds on complete hyperbolic complex manifolds and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence, we obtain a univalent solution (f (t) : M -> N) of any Loewner-Kufarev PDE. The problem of finding solutions given by univalent mappings (f (t) : M -> a", (n) ) is reduced to investigating whether the complex manifold a(a) (ta parts per thousand yen0) f (t) (M) is biholomorphic to a domain in a", (n) . We apply such results to the study of univalent mappings f: B (n) -> a", (n)