Automorphisms of affine Veronese surfaces

We prove that every derivation and every locally nilpotent derivation of the subalgebra $K[x^n,x^{n-1}y,\dots ,x{y}^{n-1},y^n]$, where $n\ge 2$, of the polynomial algebra $K[x,y]$ in two variables over a field $K$ of characteristic zero is induced by a derivation and a locally nilpotent derivation of $K[x,y]$, respectively. Moreover, we prove that every automorphism of $K[x^n,x^{n-1}y,\dots ,x{y}^{n-1},y^n]$ over an algebraically closed field $K$ of characteristic zero is induced by an automorphism of $K[x,y]$. We also show that the group of automorphisms of $K[x^n,x^{n-1}y,\dots ,x{y}^{n-1},y^n]$ admits an amalgamated free product structure.