The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh-Nagumo equation

Symmetry reductions and exact solutions of a class of nonlinear heat equations

“…Nucci and Clarkson [71] (see also [21]) demonstrated that the symmetry reduction (5.14) of the Fitzhugh-Nagumo equation (1.9), is obtainable using the nonclassical method due to Bluman and Cole [9], though not using the direct method due to Clarkson and Kruskal [20]. Hence the nonclassical method is more general than the direct method, at least as it was originally formulated.…”

“…The condition that 27d + 2b 3 − 9bc = 0 is the necessary and sufficient condition on f (u) = u 3 + bu 2 + cu + d for the existence of nonclassical symmetry reductions of (1.1) to odes solvable in terms of elliptic functions. Furthermore, Nucci and Clarkson [71] show that elliptic functions solutions exist for the Fitzhugh-Nagumo equation (1.9) if either a = −1, a = 1 2 or a = 2, which are precisely those the three cases when the distances from the outer roots to the central one are equal.…”

“…Travelling wave solutions of the Fitzhugh-Nagumo equation (1.9) have been studied by several authors [6,7,28,42,96]. Exact solutions of (1.9) have been obtained using various techniques including Vorob'ev [97] (who calls the associated symmetry a "partial symmetry of the first type"), by Kawahara and Tanaka [55] using Hirota's bi-linear method [50], by Hereman [45] using the truncated Painlevé expansion method (see also [17]) and more recently by Nucci and Clarkson [71] using the nonclassical method (see also [21]). Exact solutions of the Huxley equation…”

“…Following Nucci and Clarkson [71], this is solvable as follow. Using (5.5) to eliminate u t in (5.1) yields …”

Some Remarks on Exp-Function Method and Its Applications-A Supplement

“…The exact solution (40) (see Ref. [22]) can also be obtained with a nonclassical method of symmetry reduction [23] or via Hirota bilinear method. [24] …”