Discretionary appointment of counsel

During a proceeding involving the dissolution of marriage, legal separation, or annulment of a minor child's parents or guardians:

(a) The court may appoint counsel or a guardian ad litem for any minor child or children of either or both parties at any time after the return day of a complaint under section 46b-45, if the court deems it to be in the best interests of the child or children. The court may appoint counsel or a guardian ad litem on its own motion, or at the request of either of the parties or of the legal guardian of any child or at the request of any child who is of sufficient age and capable of making an intelligent request.

(b) Counsel or a guardian ad litem for the minor child or children may also be appointed on the motion of the court or on the request of any person enumerated in subsection (a) of this section in any case before the court when the court finds that the custody, care, education, visitation or support of a minor child is in actual controversy, provided the court may make any order regarding a matter in controversy prior to the appointment of counsel or a guardian ad litem where it finds immediate action necessary in the best interests of any child.

(c) In the absence of an agreement of the parties to the appointment of counsel or a guardian ad litem for a minor child in the parties’ matter and a canvassing by the court concerning the terms of such agreement, the court shall only appoint such counsel or guardian ad litem under this section when, in the court’s discretion, reasonable options and efforts to resolve a dispute of the parties concerning the custody, care, education, visitation or support of a minor child have been made.

(d) If the court deems the appointment of counsel or a guardian ad litem for any minor child or children to be in the best interests of the child or children, such appointment shall be made in accordance with the provisions of section 46b-12.

Conn. Gen. Stat. Ann. § 46b-54.

The high court has said that "in the absence of strong countervailing considerations such as physical urgency or financial stringency, the better course is to appoint independent counsel whenever the issue of child custody is seriously contested."  Yontef v. Yontef, 185 Conn. 275, 284; 440 A.2d 899, 904 (Conn. 1981).