This paper is concerned with transformations between different linear damped vibrating systems which preserve the dynamic characteristics of those systems. The general linear damped vibrating system is represented by three system matrices, {K, D, M}. Every other system, {K', D', M'}, sharing the same dynamic characteristics is related to the original system by a transformation called a structure preserving transformation. By considering structure-preserving transformations which are infinitesimal modifications of the identity transformation, it is possible to develop differential equations governing the general trajectory of systems {K(sigma), D(sigma), M(sigma)} all possessing the same dynamic characteristics where sigma is a scalar parameter. Any such continuous trajectory of systems is called an isospectral flow. At any one particular value of the scalar parameter, sigma, the direction of the isospectral flow for vibrating systems can be decomposed into two parts conventional and unconventional with the former being very well understood already from the context of first-order systems and the generalised eigenvalue problem. The unconventional parts of the isospectral flows have many fascinating characteristics and are, as yet, very little understood. This paper introduces pure unconventional flows and exposes some knowledge of their behaviour.