Infinite loops of Rebirths

If a MAN don't even care to try out to attain enlightenment then he is not a prisoner.

As for the end of this "birth" cycle, it is when the mind stops conceiving "self".

Refer to SN 22.99 Gaddula Sutta.

Although I am no monk or Buddhist expert, I consider myself a Buddhist. As it is understood by me, Enlightenment CAN include the cessation of the rebirth cycle if an enlightened being so chooses it. Some do not, such as the Dahli Lama. His Inner Buddha Nature/Perfect Light, whatever you choose to call an enlightened soul, seems to have the choice of whether to continue or stop the rebirth cycle. Enlightenment seems to me to have the most to do with no longer possessing the self or I and no longer considering things from the position of a "personal singularity" or self. An enlightened being no longer need consider anything from the position of self or I, but of an all empathetic and compassionate entity no longer bound by the restrictions or "chains," if you will, of a single being. this is purely my opinion/understanding and not taken from any writings or teachings that I know of to exist. Namaste, Brian Burke Jr.

One who need to stop this cycle of birth,at least have to become sovan,A person who became to sovan state has just 7 rebirths.. All others haven't even understood sovan state'll be in this circle of rebirth

The process of purification of mind, is like the refinement of gold. One needs to do away with the impurities (fetters/hindrances) to obtain pure gold. As long as they exist, one is confined to Samsara.

According to Theravada Buddhism, there are Three Gateways to Liberation. In the Visuddhimagga, they are described as realizing the Three Characteristics of Existence.

The Triple Gateway To Liberation

It enters upon the state of the triple gateway to liberation now with the predominance of [one of] three faculties according as the contemplation occurs in [one of] the three ways. For it is the three contemplations that are called the three gateways to liberation, according as it is said: “But these three gateways to liberation lead to the outlet from the world, [that is to say,]

(i) to the seeing of all formations as limited and circumscribed and to the entering of consciousness into the signless element,

(ii) to the stirring up of the mind with respect to all formations and to the entering of consciousness into the desireless element,

(iii) to the seeing of all things (dhamma) as alien and to the entering of consciousness into the voidness element. These three gateways to liberation lead to the outlet from the world”.

-- Visuddhimagga: The Path of Purification, p. 685-88, by Ven. Buddhaghosa

A person who doesn't exert himself to attain it will obviously go without an attainment of knowledge & vision leading to the non emergence of a future due to an exhaustion of it's requisite causes & conditions.

The cessation of that which changes as it persists is thought about as it's cessation principle on account of which a yogi realizes the foremost meditative release.

The knowledge & vision of the release arises when the mind is properly inclined to it through the development of perceptions leading to disenchantment with transient elements as opposed to an inclination to a perpetual conception & perception of & through doors of the senses on account of fascination & enchantment with various states of perception & feeling.

The cessation is a freedom to be directly realized, it is a freedom from being and a freedom to be and this seeing with wisdom [judgement] destroys defilements of greed, anger & delusion as that sorrowless state is void of these elements.

One who practices this release of the mind will eventually remove delusion, greed & anger from his mind altogether, destroying the root [of becoming] they say, this is called unprovoked release or nibbana with residue. Their minds are in that made like the cessation element void of delusion and their incline to cessation as they become completely cool in regards to expectations & determinations for a future.

With their mind having no inclination to conception & perception of this or that state of existence due to disenchantment and with the breakup of the body by which they were experiencing pleasant & unpleasant sensations all modes of being come to an end.

To your first question, the Pacceka-Buddha are individuals who attained enlightenment without the Buddha's teaching. They are very, very rare. Since they achieved enlightenment solely on their own, there is no way to know if they cared about enlightenment; they might have, and they might have not, but it does not really matter.

To your second question, 'Enlightenment' is the only way to end the birth-cycle.

Vimalakirti Sutra:

Śāriputra: Goddess, where will you be born when you transmigrate after death?

Goddess: I will be born where all the magical incarnations of the Tathāgata are born.

Śāriputra: But the emanated incarnations of the Tathāgata do not transmigrate nor are they born.

Goddess: All things and living beings are just the same; they do not transmigrate nor are they born!

Śāriputra: Goddess, how soon will you attain the perfect enlightenment of buddhahood?

Goddess: At such time as you, elder, become endowed once more with the qualities of an ordinary individual, then will I attain the perfect enlightenment of buddhahood.

Śāriputra: Goddess, it is impossible that I should become endowed once more with the qualities of an ordinary individual.

Goddess: Just so, reverend Śāriputra, it is impossible that I should attain the perfect enlightenment of buddhahood! Why? Because perfect enlightenment stands upon the impossible. Because it is impossible, no one attains the perfect enlightenment of buddhahood.

Śāriputra: But the Tathāgata has declared: “The tathāgatas, who are as numerous as the sands of the Ganges, have attained perfect buddhahood, are attaining perfect buddhahood, and will go on attaining perfect buddhahood.”

Goddess: Reverend Śāriputra, the expression “the buddhas of the past, present, and future” is a conventional expression made up of a certain number of syllables. The buddhas are neither past, nor present, nor future. Their enlightenment transcends the three times! But tell me, elder, have you attained the state of arhat?

Śāriputra: It is attained, because there is no attainment.

Goddess: Just so, there is perfect enlightenment because there is no attainment of perfect enlightenment.

Enlightenment(Nibbana) is the only way out. There is no other. As long as the causes are present, the effect will arise.