A partial differential approximant (or an ordinary rational approximant) which approximates a function, f ( x , y ), of two variables may be invariant under certain changes of variable ( x , y ) ⇒ ( x ¯ , y ¯ ). We prove that, under fairly mild conditions, such invariance is possible for only five classes of variable transformations, namely: the Euler transformations x ¯ = A x /(1+B x ) and y ¯ = C y /(1+D y ); homogeneous linear transformations of x and y ; Euler-rotations’ x ¯ = (A 1 x +A 2 y )/ (1+C x +D y ) and y ¯ = (B 1 x +B 2 y )/(1+C x +D y ); the skew Euler transformations x ¯ = A y /(1+B y ) and y ¯ = C x /(1+D x ); and transformations such as x ¯ = A x and y ¯ = x C( x )+B y , where C( x ) is a polynomial. This result also holds for ordinary two variable rotational approximants.