When there is found in any town in this state any child of sound mind who is physically disabled or who is afflicted with poliomyelitis or rheumatic fever, or any uncontagious disabling disease, and who is unable to pay and whose relatives who are legally liable for his support are unable to pay the full cost of treating such disease, if such child and one of such relatives reside in this state, the selectmen of such town, or the guardian or any relative of such child, or any public health agency, physician, physician assistant or advanced practice registered nurse in this state, may make application to The Children’s Center, located at Hamden, for the admission of such child to said center. Said center shall admit such child if such child is pronounced by a physician, a physician assistant or an advanced practice registered nurse on the staff of said center, after examination, to be fit for admission, and said center shall keep and support such child for such length of time as it deems proper. Said center shall not be required to admit any such child unless it can conveniently receive and care for him at the time such application is made, and said center may return to the town in which such child resides any child so taken who is pronounced by a physician, a physician assistant or an advanced practice registered nurse on the staff of said center, after examination, to be unfit for retention, or who, by reason of improvement in his condition or completion of his treatment or training, ought not to be further retained. The center may refuse to admit any child who is pronounced by a physician, a physician assistant or an advanced practice registered nurse on the staff of said center, after examination, to be unfit for admission, and may refuse to admit any such child when the facilities at the center will not, in the judgment of such physician, physician assistant or advanced practice registered nurse, permit the center to care for such child adequately and properly.