Standard Morse lemma states that if the singularity of a smooth function $f$ is non-degenerate, one can choose coordinates such that function has a "quadratic" form $f = \sum \limits_i x_i^2 - \sum \limits_j x_j^2$. To prove this one uses an integral representation $f(x) - f(0) = \int \limits_0^1 \frac{d}{dt}f(tx) dt$ to get a quadratic form with in general non-constant coefficients $f(x) = \sum \limits_{ij} h_{ij} (x) x_i x_j$.
Now suppose that we have two functions $f$ and $g$ of two variables. We know that $g$ has a non-degenerate minimum at origin, while $f$ does not have any singularities. When can we choose coordinates such that $f = \alpha x, g = x^2 + y^2$, where $\alpha$ is constant?