Let the invariant integral:

and the Lie point symmetry:

The invariance condition is:

that is this symmetry leaves invariant the variational integral under investigation.

Let the Lie point symmetry

and the variational integral

The invariance condition is:

here we can see that for the symmetry to keep invariant the variational integral defined by

[x, t] (u_{, t}-u_{, xx}) the relation between the constants

_{3}+_{4}+_{5}=0 must hold.

Let the Lie-bäcklund symmetry

and the variational integral

The invariance condition is:

the above system must hold for every value of the independent variables, the dependent ones and their derivatives. Thus obtaining a system for the constants.