The generalized Cartan-Monge type approach to the characteristics method is discussed from the geometric point of view. Its application to the classical one-dimensional linear heat equation $u_t-u_{xx}=0$ is presented. It is shown that the corresponding exact solution of the Cauchy problem can be represented in a classical functional-analytic Gauss type form.
An approach based on the spectral and Lie - algebraic techniques for constructing vertex operator representation for solutions to a Riemann type hydrodynamical hierarchy is devised. A functional representation generating an infinite hierarchy of dispersive Lax type integrable flows is obtained.
