The ‘separant’ of the evolution equation ut = F, where F is some differentiable function of the derivatives of u up to order m, is the partial derivative where . As an integrability test, we use the formal symmetry method of Mikhailov–Shabat–Sokolov, which is based on the existence of a recursion operator as a formal series. The solvability of its coefficients in the class of local functions gives a sequence of conservation laws, called the ‘conserved densities’ . We apply this method to the classification of scalar evolution equations of orders , for which and are non-trivial, i.e. they are not total derivatives and is not linear in its highest order derivative. We obtain the ‘top level’ parts of these equations and their ‘top dependencies’ with respect to the ‘level grading’, that we defined in a previous paper, as a grading on the algebra of polynomials generated by the derivatives ub+i, over the ring of functions of . In this setting b and i are called ‘base’ and ‘level’, respectively. We solve the conserved density conditions to show that if depends on then, these equations are level homogeneous polynomials in , . Furthermore, we prove that if is non-trivial, then , with while if is trivial, then , where and α, β, γ, λ and μ are functions of . We show that the equations that we obtain form commuting flows and we construct their recursion operators that are respectively of orders 2 and 6 for non-trivial and trivial respectively. Omitting lower order dependencies, we show that equations with non-trivial and b = 3 are symmetries of the ‘essentially non-linear third order equation’; for trivial , the equations with b = 5 are symmetries of a non-quasilinear fifth order equation obtained in previous work, while for b = 3, 4 they are symmetries of quasilinear fifth order equations.