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To search for inequivalent group invariant solutions, a general andsystematic approach is established to construct two-dimensional optimalsystems, which is based on commutator relations, adjoint matrix and theinvariants. The details of computing all the invariants for two-dimensionalsubalgebras is presented and the optimality of twodimensional optimal systemsis shown clearly under different values of invariants, with no further proof.Applying the algorithm to (1+1)-dimensional heat equation and (2+1)-dimensionalNavier-Stokes (NS) equation, their twodimensional optimal systems are obtained,respectively. For the heat equation, eleven two-parameter elements in theoptimal system are found one by one, which are discovered more comprehensive.The two-dimensional optimal system of NS equations is used to generateintrinsically different reduced ordinary differential equations and someinteresting explicit solutions are provided.

The construction of two-dimensional optimal systems for the invariant solutions