Equivalence classes and linearization of the Riccati and Abel chain

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The problem of linearization by point transformations is solved for equations in the generalized Riccati and Abel chain of order not exceeding the fourth. It is shown in particular that nonlinear third order and fourth order equations from the chain are not linearizable by any point transformations. The Lie pseudo-group of equivalence transformations for equations of arbitrary orders from the chain are then found, together with expressions for the transformed parameter functions. An important subgroup of the group of equivalence transformations found is considered and some associated equivalence classes are exhibited.