The map $\phi(x,y)=(\sqrt{1+x^2}-y,x)$ of the plane is area preserving and has the remarkable property that in numerical studies it shows exact integrability: The plane is a union of smooth, disjoint, invariant curves of the map $\phi$. However, the integral has not explicitly been known. In the current paper we will show that the map $\phi$ does not have an algebraic integral, {i.e.,} there is no non-constant function $F(x,y)$ such that \begin{enumerate} \item $F\circ\phi=F$; \item There exists a polynomial $G(x,y,z)$ of three variables with %and \[G(x,y,F(x,y))=0.\] \end{enumerate} Thus, the integral of $\phi$, if it does exist, will have complicated singularities. We also argue that if there is an analytic integral $F$, then there would be a dense set of its level curves which are algebraic, and an uncountable and dense set of its level curves which are not algebraic.