Equivalence and symmetries of second-order differential equations

In this article we investigate the equivalence of underdetermined differential equations and differential equations with deviations of second order with respect to the pseudogroup of transformations $$ \bar x $$ = φ(x), ȳ = ȳ($$ \bar x $$) = L(x) + y(x), $$ \bar z $$ = $$ \bar z $$($$ \bar x $$) = M(x) + z(x). Our main aim is to determine such equations that admit a large pseudogroup of symmetries. Instead the common direct calculations, we use some more advanced tools from differential geometry, however, our exposition is self-contained and only the most fundamental properties of differential forms are employed.