ExecuteInEditMode Component construct "twice" and "reset" custom classes

Hello community. I have an independant class, and I want, from Editor, create a component with a member in this type. It works well, but when I click on "Play Button", custom classes AND private members are reset, like the RAM were flushed by garbage collector of something. I manage to reproduce this behaviour in an EMPTY project, just add this two script files:

In a MyClass.cs :

 #if UNITY_3_3
 using UnityEngine;
 using UnityEditor;
 #endif

 public class MyClass
 {
     public string Name;

     #if UNITY_3_0 || UNITY_3_1 || UNITY_3_2 || UNITY_3_3
     [MenuItem("Custom/InitComponent")]
     static public void MInitComponent()
     {
         GameObject MyCubeGO = GameObject.CreatePrimitive(PrimitiveType.Cube);

         MyComponent myComp = MyCubeGO.AddComponent<MyComponent>() as MyComponent;
         myComp.MyPublicBool= true;
         myComp.MyClassInstance.Name = "John Doe";
     }
     #endif

}

And in a MyComponent.cs :

 using System;
 using System.Collections;


 #if UNITY_3_0 || UNITY_3_1 || UNITY_3_2 || UNITY_3_3
 using UnityEngine;

 [ExecuteInEditMode()]
 public class MyComponent : MonoBehaviour 
 {
     private bool MyPrivateBool;
     public bool MyPublicBool;
     public MyClass MyClassInstance;

     public bool AlreadyConstructed=false;

 // Use this for initialization
 void Awake() 
 {
     if (!AlreadyConstructed)
     {
         Debug.Log("Awaking...");
         MyClassInstance = new MyClass();
         MyPrivateBool = true;
         MyPublicBool = true;

         AlreadyConstructed = true;
     }
 }
 
 // Update is called once per frame
 void Start() 
 {
     Debug.Log("Starting...");
     Debug.Log("MyPublicBool is at value =" + MyPublicBool);
     Debug.Log("MyPrivateBool is at value =" + MyPrivateBool);// <==== OK(true) In  Editor, RESET (at false) when cliking on PLAY !

     Debug.Log("My Name is ="+MyClassInstance.Name); // <==== OK In Editor, CRASH when cliking on PLAY !
     MyPrivateBool = false; //Checking the reset here too.
 }
 }
 #endif

As a result , I have for output :

Awaking... Starting... MyPublicBool is at value =True MyPrivateBool is at value =True My Name is =John Doe

//here I click on PLAY Starting... MyPublicBool is at value =True MyPrivateBool is at value =False //WHICH IS WRONG NullReferenceException: Object reference not set to an instance of an object at MyComponent.Start () [0x0003e] in C:\UNITY\Debug01\Assets\MyComponent.cs:37

(Filename: Assets/MyComponent.cs Line: 37) // it this line: Debug.Log("My Name is ="+MyClassInstance.Name);

How can I avoid this kind of behaviour, to keep my initialisation also when I click on "PLAY" ? Thank you for your help!