The meaning of the word eutony is well-balanced tension.
Gerda Alexander, developer of this somatic practice, said that it is a Western way of experiencing unity of the whole person. She also said this unity frees your creative forces and develops your ability to make contact with others while maintaining your individuality.
The goal of this practice is to widen and deepen your whole range of expression and open up your creativity. The work is gentle and never forces. Presence is of the upmost importance:
This form of somatic education is not only about movement awareness but also about developing depth in yourself and growing toward being a total person.
Here's an interview with Gerda Alexander from the January/February 1986 issue of Yoga Journal.
"Interview with Gerda Alexander" by David Bersin in Bone, Breath, and Gesture: Practices of Embodiment, edited by Don Hanlon Johnson.
Discovering the Body's Wisdom by Mirka Knaster has a chapter on this somatic practice.
Eutony: The Holistic Discovery of the Total Person by Gerda Alexander.
Planet Medicine: Modalities by Richard Grossinger has a chapter on this practice.
Videos on YouTube (In Spanish, but some of the visual information is enlightening, even if you don't understand Spanish.)
Thinking Through the Body provides a history and description of this practice.