Let $\mathcal{P}$ be a convex polygon with $n$ sides, where $n \geq 3$. We are interested in finding all possible values of $n$ such that there exists a triangulation of $\mathcal{P}$ into only isosceles triangles when $\mathcal{P}$ is regular.

A triangulation of $\mathcal{P}$ means choosing $n-3$ diagonals such that the polygon is divided into $n-2$ triangles. If all these triangles are isosceles then each angle of these triangles must be a rational multiple of $\pi$ due to the polygon being regular and having equal angles.

### Conditions for Isosceles Triangles

For a regular polygon, each internal angle is given by:

$$\theta = \frac{(n-2)\pi}{n}$$

Consider the condition for the isosceles triangle inside $\mathcal{P}$. For a triangle having two equal angles, say $\alpha$, we know:

$$2\alpha + \beta = \pi$$

Thus, $\alpha$ must be in the form of $\frac{k\pi}{n}$ for some integer $k$.

### Constructing Isosceles Triangles in the Polygon

To have a triangulation only with isosceles triangles, the base angles of each triangle and the central angle at the vertex of the polygon opposite this base must also conform to being a rational fraction of $\pi/n$.

This condition implies that:

- The central angles, which are subtended by two consecutive vertices forming a triangle, should be of the form $\frac{2m\pi}{n}$
- The other two angles must be equal and $\frac{k\pi}{n}$.

Important relationships can be derived based on symmetry, requiring $n$ to be such that:

1. There is an even lattice of division within $\pi$.
2. The internal angles from the diagonals can resonate across all such constructions of isosceles triangles.

### Number Theoretic Characterization

It is required that $n$ is formed such that there's symmetry allowing for triangulations into isosceles triangles. Classical constructions indicate these numbers satisfy the property of Steuerwald's theorem or Maurer's theorem which are related to number theoretic solutions dealing with cyclotomic fields.

The critical component is that $n$ must possess properties of having a power of two times one more than a power of two, formally:

$$n = 2^a(2^b + 1)$$

where $a$ and $b$ are nonnegative integers and not simultaneously zero. This condition ensures that we can indeed partition the structure congruently into isosceles triangles internal to the polygon.

### Conclusion

Hence, the possible values of $n$ that allow for such triangulation into only isosceles triangles of a regular polygon are:

$$\boxed{n = 2^a (2^b + 1) \text{ where } a, b \text{ are nonnegative integers not both zero.}}$$