**Conservation laws, potential symmetries and Darboux transformation
of linear parabolic equations**

**Abstract:**

Extended investigation on conservation laws and potential
symmetries is performed for the class of linear (1+1)-dimensional
second-order parabolic equations. The group classification of this
class is revised with usage of admissible transformations, the
notion of normalized classes of differential equations and the
adjoint variational principle. All possible potential conservation
laws are described completely. They are exhausted by local ones. For
any equation under consideration the characteristic space of local
conservation laws is isomorphic to the solution set of the adjoint
equation. Effective criteria on existence of potential symmetries
are proposed. Their proofs involves a sophisticated technique based
on different representations of potential systems, the notion of a
potential equation associated with a tuple of characteristics,
prolongation of the equivalence group to the whole potential frame
and application of multiple dual Darboux transformations. The
simplest potential symmetries of the linear heat equation, which are
associated with single conservation laws, are classified with
respect to its point symmetry group. Equations possessing infinite
series of potential symmetry algebras are studied in detail.

Joint work with *Michael Kunzinger* (University of Vienna, Austria) and
*Nataliya Ivanova* (Institute of Mathematics of NAS of Ukraine, Kyiv,
Ukraine).