  In this work, we construct the general solution to the Heat Equation (HE) and
to many tensor structures associated to the Heat Equation, such as Symmetries,
Lagrangians, Poisson Brackets (PB) and Lagrange Brackets, using newly devised
techniques that may be applied to any linear equation (e.g., Schroedinger
Equation in field theory, or the small-oscillations problem in mechanics). In
particular, we improve a time-independent PB found recently which defines a
Hamiltonian Structure for the HE, and we construct an Action Principle for the
HE. We also find a new structure, which we call a Metric Structure, which may
be used to define alternative anti-commutative "Hamiltonian" theories, in which
the Metric- or M-Hamiltonians have to be explicitly time-dependent. Finally, we
map some of these results to the Potential Burgers Equation.
