# Rational Maps with Invariant Surfaces

## TL;DR

This paper introduces new integrable rational maps in four dimensions with two invariants, revealing unexpected geometric behaviors and enabling reconstruction of the maps from invariants, expanding understanding of such systems.

## Contribution

It presents novel examples of integrable rational maps with unique geometric properties and methods for reconstructing maps from invariants, surpassing previous classifications.

## Key findings

- Orbits confined to non algebraic varieties
- Reconstruction of maps from invariants
- Discovery of non trivial fibrations of initial conditions space

## Abstract

We provide new examples of integrable rational maps in four dimensions with two rational invariants, which have unexpected geometric properties, as for example orbits confined to non algebraic varieties, and fall outside classes studied by earlier authors. We can reconstruct the map from both invariants. One of the invariants defines the map unambiguously, while the other invariant also defines a new map leading to non trivial fibrations of the space of initial conditions.