Advise the task management system (internal and from outside) for small-medium businesses and personal use

I would like to implement a task management system, you need something like this. Tell me if there is already such or it is necessary to sit down and to write a Bicycle? Preferably in PHP (to set a simple shared hosting), but can be any other language if it's worth it.

Thanks in advance.

Here that approximately it takes at the points:

Each task (ticket) may:

• come in the form of requests from client (mail and/or form on the public site without a login)
• or to be created inside

to belong to (respectively should be visible only to those who should)

• all registered users
• or workgroup
• or someone in person

• consist of subtasks
• and/or be a subtask of another task

have pratasenya text/links (WiKi?) and files.

to have or not have

• status
• the scheduled start time
• and the end of the run
• a critical deadline
• responsible
• and involved entities.