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To obey classical Lie-symmetries, differential equations
for unknown functions 
of independent variables 
must be forminvariant against infinitesimal transformations
in first order of 
To transform the equations (1)
by (2), derivatives of 
must be transformed, i.e. the part
linear in 
must be determined. The corresponding formulas are
(see e.g. [10], [20])
where 
means total differentiation w.r.t. 
and
from now on lower latin indices of functions 
(and later 
)
denote partial differentiation w.r.t. the independent variables 
(and later 
).
The complete symmetry condition then takes the form
where mod 
means that the original PDE-system is used to replace
some partial derivatives of 
to reduce the number of independent
variables, because the symmetry condition (4) must be
fulfilled identically in 
and all partial
derivatives of 
For point symmetries, 
are functions of 
and for contact symmetries they depend on 
and
We restrict ourself to point symmetries as those are the only
ones that can be applied by the current version of the program APPLYSYM
(see below). For literature about generalized symmetries see [1].
Though the formulation of the symmetry conditions (4),
(5), (3)
is straightforward and handled in principle by all related
programs [1], the computational effort to formulate
the conditions (4) may cause problems if
the number of 
and 
is high. This can
partially be avoided if at first only a few conditions are formulated
and solved such that the remaining ones are much shorter and quicker to
formulate.
A first step in this direction is to investigate one PDE 
after another, as done in [22]. Two methods to partition the
conditions for a single PDE are described by Bocharov/Bronstein
[9] and Stephani [20].
In the first method only those terms of the symmetry condition
are calculated which contain
at least a derivative of 
of a minimal order 
Setting coefficients
of these u-derivatives to zero provides symmetry conditions. Lowering the
minimal order m successively then gradually provides all symmetry conditions.
The second method is even more selective. If 
is of order n
then only terms of the symmetry condition 
are generated which
contain 
th order derivatives of 
Furthermore these derivatives
must not occur in 
itself. They can therefore occur
in the symmetry condition
(4) only in
i.e. in the terms
If only coefficients of 
th order derivatives of 
need to be
accurate to formulate preliminary conditions
then from the total derivatives to be taken in
(3) only that part is performed which differentiates w.r.t. the
highest 
-derivatives.
This means, for example, to form only
if the expression, which is to be differentiated totally w.r.t. 
,
contains at most second order derivatives of 
The second method is applied in LIEPDE.
Already the formulation of the remaining conditions is speeded up
considerably through this iteration process. These methods can be applied if
systems of DEs or single PDEs of at least second order are investigated
concerning symmetries.
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