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? 



Modification of Morse lemma with two functions
In general, this is not possible. One needs a neighbourhood $U$ of the origin such that on every level set $f^{-1}(c)\cap U$, the minimum of $g|\_U$ is exactly $(c/\alpha)^2$. If this condition is satisfied, then regard $g$ as a fibrewise Morse function for the family $f\colon U\to V\subset\mathbb R$. Because the Morse lemma holds in families, one can now prove the claim.


